problems Open of Paul Erds in graph theory o
F. R. K. Chung University of Pennsylvania Philadelphia, Pennsylvania 19104
The main treasure that Paul Erds has left us is his collection of problems, o most of which are still open today. These problems are seeds that Paul sowed and watered by giving numerous talks at meetings big and small, near and far. In the past, his problems have spawned many areas in graph theory and beyond (e.g., in number theory, probability, geometry, algorithms and complexity theory). Solutions or partial solutions to Erds problems usually lead to further o questions, often in new directions. These problems provide inspiration and serve as a common focus for all graph theorists. Through the problems, the legacy of Paul Erds continues (particularly if solving one of these problems results in o creating three new problems, for example.) There is a huge literature of almost 1500 papers written by Erds and his o (more than 460) collaborators. Paul wrote many problem papers, some of which appeared in various (really hard-to-find) proceedings. Here is an attempt to collect and organize these problems in the area of graph theory. The list here is by no means complete or exhaustive. Our goal is to state the problems, locate the sources, and provide the references related to these problems. We will include the earliest and latest known references without covering the entire history of the problems because of space limitations. (The most up-to-date list of Erds' papers can be found in [65]; an electronic file is maintained by Jerry o Grossman at [e-mail].) There are many survey papers on the impact of Paul's work, e.g., see those in the books: "A Tribute to Paul Erds" o [84], "Combinatorics, Paul Erds is Eighty", Volumes 1 and 2 [83], and "The o Mathematics of Paul Erds", Volumes I and II [81]. o To honestly follow the unique style of Paul Erds, we will mention the fact o that Erds often offered monetary awards for solutions to a number of his fao vorite problems. In November 1996, a committee of Erds' friends decided no o more such awards will be given in Erds' name. However, the author, with the o help of Ron Graham, will honor future claims on the problems in this paper, provided the requirements previously set by Paul are satisfied (e.g., proofs have been verified and published in recognized journals).
To
appear in Journal of Graph Theory c 1997 John Wiley and Sons, Inc.
1
Throughout this paper, the constants c, c , c1 , c2 , . . . and extremal functions f (n), f (n, k), f (n, k, r, t), g(n), . . . are used extensively, although within the context of each problem, the notation is consistent. We interpret graph theory in the broad sense, for example, including hypergraphs and infinite graphs.
Ramsey theory
For two graphs G and H, let r(G, H) denote the smallest integer m satisfying the property that if the edges of the complete graph Km are colored in blue and red, then there is either a subgraph isomorphic to G with all blue edges or a subgraph isomorphic to H with all red edges 1 . The classical Ramsey numbers are those for the complete graphs and are denoted by r(s, t) = r(Ks , Kt ). Classical Ramsey numbers In 1935, Erds and Szekeres [135] gave an upper bound for the Ramsey o number r(s, t). In 1947, Erds [90] used probabilistic methods to establish a o lower bound for r(n, n). The following results play an essential role in laying the foundations for both Ramsey theory and combinatorial probabilistic methods: 1 (1 + o(1)) n2n/2 < r(n, n) e 2 2n - 2 n-1 (1)
In the past fifty years, relatively little progress has been made. The current best lower bound and upper bound are due to Spencer [209] and Thomason [211], respectively. 2n - 2 2 n/2 n2 < r(n, n) < n-1/2+c/ log n (1 + o(1)) n-1 e (1) Conjecture, 1947 ($100) The following limit exists:
n
lim r(n, n)1/n = c
(2) Problem, 1947 ($250) Determine the value of the limit c above (if it exists). By (1), the limit, if it exists, is between 2 and 4. The proof for the lower bound in (1) is by the probabilistic method. (3) A problem on explicit constructions ($100) Give a constructive proof for r(n, n) > (1 + c)n for some constant c > 0. The best known constructive lower bound nc log n/ log log n is due to Frankl and Wilson [144].
1 In
this paper, by a graph we mean a simple loopless graph, unless otherwise specified.
2
(4) Conjecture, 1947 For fixed l, r(k, l) > k l-1 (log k)cl
for a suitable constant cl > 0 and k sufficiently large. For r(3, n), Kim [163] recently used a complicated probabilistic argument to prove the following lower bound for r(3, n) which is of the same order as the upper bound previously established by Ajtai, Komls and Szemerdi o e [2]: c2 n 2 c1 n 2 < r(3, n) < . log n log n (5) It would be of interest to have an asymptotic formula for r(3, n). For r(4, n) the best lower bound is c(n/ log n)5/2 due to Spencer [207] and the upper bound is c n3 / log2 n, proved by Ajtai, Komls and Szemerdi o e [2]. (6) Problem $250 (see [78]) Prove or disprove that r(4, n) > for n sufficiently large. (7) Conjecture (proposed by Burr and Erds [80]) o r(n + 1, n) > (1 + c)r(n, n) for some fixed c > 0. (8) Conjecture (proposed by Erds and Ss [80]) o o r(n + 1, 3) - r(n, 3) In particular, prove or disprove (9) r(n + 1, 3) - r(n, 3) 0. n Graph Ramsey numbers (10) A conjecture on Ramsey number for bounded degree graphs (proposed by Burr and Erds [35]) o For every graph H on n vertices in which every subgraph has minimum degree c, r(H, H) c n 3
n3 logc n
where the constant c depends only on c. We remark that Chvtal, Rdl, Szemerdi and Trotter [57] proved that for a o e graphs with bounded maximum degree, the Ramsey number grows linearly with the size of the graph. An extension of this result was obtained by Chen and Schelp [44] by showing that the bounded degree condition can be replaced by a somewhat weaker requirement. In particular, they proved that the Ramsey number for a planar graph on n vertices is bounded above by cn. Rdl and Thomas [194], generalizing results in [44], showed that o graphs with bounded genus have linear Ramsey numbers. (11) Conjecture (proposed by Erds and Graham [102]) o If G has n edges for n 4, then 2 r(G, G) r(n, n). (12) More generally, if G has
n 2
+ t edges, then r(G, G) r(H, H)
where H denotes the graph formed by connecting a new vertex to t of the vertices of a Kn and t n. (13) Problem [72] Is it true that if a graph G has e edges, then r(G, G) < 2ce for some constant c? (14) A problem on n-chromatic graphs [82] Let H denote an n-chromatic graph. Is it true that r(H, H) > (1 - )n r(n, n) holds for any 0 < < 1 provided n is large enough? (15) Problem [82] Prove that there is some number n, > 0 so that for all n and all H of chromatic r(H, H) > . r(n, n) This is a modified version of an old conjecture r(H, H) r(n, n) (see [28]) which, however, has a counterexample for n = 4 given by Faudree and McKay [138] who showed r(W, W ) = 17 for the pentagonal wheel W . (16) Conjecture [63] For some > 0,
1/2
r(C4 , Kn ) = o(n2- ). 4
It is known that c( n 2 n 3/2 ) > r(C4 , Kn ) > c( ) log n log n
where the lower bound is proved by probabilistic methods [207] and the upper bound is due to Szemerdi (unpublished, also see [95]). e (17) Problem (proposed by Erds, Faudree, Rousseau and Schelp [95]) o Is it true that r(Cm , Kn ) = (m - 1)(n - 1) + 1 if m n > 3? The answer is affirmative if m n2 - 2 (see [27, 139]). (18) Problem (proposed by Burr, Erds, Faudree, Rousseau and Schelp [37]) o Determine r(C4 , K1,n ). It is known that n+ n + 1 r(C4 , K1,n ) n + n - 6n11/40
where the upper bound can be easily derived from the Turn number of C4 a and the lower bound can found in [37]. Fredi can show (unpublished) be u that r(C4 , K1,n ) = n + n holds infinitely often. (19) Conjecture (proposed by Burr, Erds, Faudree, Rousseau and Schelp [37]) o If G is fixed and n is sufficiently large, then r(G, T ) r(G, K1,n-1 ) for every tree T on n vertices. (20) A Ramsey problem for n-cubes (proposed by Burr and Erds [35]) o Let Qn denote the n-cube on 2n vertices and n2n-1 edges. Prove that r(Qn , Qn ) c2n . Beck [18] showed that r(Qn , Qn ) c2n . (21) Linear Ramsey bounds (proposed by Burr, Erds, Faudree, Rousseau and Schelp [37]) o Suppose a graph G satisfies the property that every subgraph of G on p vertices has at most 2p - 3 edges. Is it true that r(G, G) cn? In general, the problem of interest is to characterize graphs whose Ramsey number r(G, G) is linear. Alon showed [6] that if no two vertices of degree exceeding two are adjacent in a graph G, then the Ramsey number of G is linear.
2
5
(22) Graphs with linear Ramsey bounds (proposed by Burr, Erds, Faudree, Rousseau and Schelp [37]) o For a graph G, where G is Q3 , K3,3 or H5 (formed by adding two vertexdisjoint chords to C5 ), is it true that r(G, H) cn for any graph H with n vertices? Multi-colored Ramsey numbers For graphs Gi , i = 1, . . . , k, let r(G1 , . . . , Gk ) denote the smallest integer m satisfying the property that if the edges of the complete graph Km are colored in k colors, then for some i, 1 i k, there is a subgraph isomorphic to Gi with all edges in the i-th color. We denote r(n1 , . . . , nk ) = r(Kn1 , . . . , Knk ). (23) Conjecture ($250, a very old problem of Erds') o Determine lim (r(3, . . . , 3))1/k .
k k
This problem goes back essentially to Schur [200] who proved r(3, . . . , 3) < e k!
k
It is known [45] that r(3, . . . , 3) is supermultiplicative so that the above limit exists.
k
(24) Problem ($100) Is the limit above finite or not? Any improvement for small values of k will give a better general lower bound. The current best lower bound is (321)1/5 using the 5-colored construction given by Exoo [134]. (25) A coloring problem for cycles (proposed by Erds and Graham [80]) o Show that
k
k
lim
r(C2n+1 , . . . , C2n+1 ) =0 r(3, . . . , 3)
k
This problem is open even for n = 2. (26) A problem on three cycles (proposed by Bondy and Erds [80]) o r(Cn , Cn , Cn ) 4n - 3. For odd n, if the above inequality is true, it is the best possible. Recently, Luczak (personal communication) showed that r(Cn , Cn , Cn ) 4n + o(n). 6
Size Ramsey numbers The size Ramsey number r (G) is the least integer m for which there exists ^ a graph H with m edges so that in any 2-coloring of the edges of H, there is always a monochromatic copy of G in H. (27) A size Ramsey problem for bounded degree graphs (proposed by Beck and Erds [189]) o For a graph G on n vertices with bounded degree, prove or disprove that r (G) cn. ^ The case for paths was proved by Beck [19] and the case for cycles was proved by Haxell, Kohayakawa, and Luczak [162]. Friedman and Pippenger [145] settled the case for any bounded degree tree. (28) A size Ramsey problem (proposed by Burr, Erds, Faudree, Rousseau and Schelp [36]) o For F1 = s K1,ni and F2 = t K1,mi , prove that i=1 i=1
s+t
r (F1 , F2 ) = ^
k=2
lk
where lk = max{ni + mj - 1 : i + j = k}. It was proved in [36] that r (sK1,n , tK1,m ) = (m + n - 1)(s + t - 1). ^ Induced Ramsey theory The induced Ramsey number r (G) is the least integer m for which there exists a graph H with m vertices so that in any 2-coloring of the edges of H, there is always an induced monochromatic copy of G in H. The existence of o o r (G) was proved independently by Deuber [60], Erds, Hajnal and Psa [111], and Rdl [193]. o (29) Problem (proposed by Erds and Rdl [77]) o o If G has n vertices, is it true that r (G) < cn for some absolute constant c? This holds for a bipartite graph [193]. Luczak and Rdl [183] showed o that a graph on n vertices with bounded degree has its induced Ramsey number bounded by a polynomial in n, confirming a conjecture of Trotter. Suppose G has k vertices and H has t k vertices. Kohayakawa, Prmel, o and Rdl [166] proved that the induced Ramsey number r (G, H) satisfies o the following bound: r (G, H) tck log q 7
where q denotes the chromatic number of H and c is some absolute constant. This implies r (G) < ncn log n .
Extremal graph theory
Turn numbers a For a graph H, let t(n, H) denote the Turn number of H, which is the a largest integer m such that there is a graph G on n vertices and m edges which does not contain H as a graph. (30) A conjecture on the Turn number for complete bipartite graphs a Prove that t(n, Kr,r ) > cn2-1/r where c is a constant depending on r (but independent of n). An upper o a o bound for t(n, Kr,r ) of the same order was proved by Kvri, Ss and Turn [170] and Erds independently. This long standing problem is well a o known as the problem of Zarankiewicz, first considered by Zarankiewicz [216] in 1951. However, it is included in the favorite problems of Erds o [79], who proposed many variations of this problem. The above conjecture is true for r = 2 and 3 (see [124]) and unsolved for r 4. The lower bound of t(n, Kr,r ) > cn2-2/(r+1) can be proved by probabilistic methods [131]. Recently, Kollr, Rnyai and Szab [167] showed that t(n, Kr,s ) > cn2-1/r a o o if r 4 and s r! + 1. (31) A conjecture on the Turn number for bipartite graphs ($100) a (proposed by Erds and Simonovits [129], 1984) o If H is a bipartite graph such that every induced subgraph has a vertex of degree r, then the Turn number for H satisfies: a t(n, H) = O(n2-1/r ). This conjecture is open even for r = 3. This problem is essentially the special case which has eluded the power of the celebrated Erds-Stone Theorem [132] and Erds-Simonovits-Stone o o Theorem [126] which can be used to determine t(n, H) asymptotically for all H with chromatic number (H) at least 3. Namely, t(n, H) = (1 - 1/((H) - 1)) n + o(n2 ). 2 A variation of the above problem is the following: (32) Conjecture (proposed by Erds and Simonovits [129], 1984) o If a bipartite graph H contains a subgraph H with minimum degree greater than r, then t(n, H) cn2-1/r+ for some > 0. 8
(33) A conjecture on the exponent of a bipartite graph (proposed by Erds and Simonovits [129], 1984) o For all rationals 1 < p/q < 2, there exists a bipartite graph G such t(n, G) = (np/q ). (34) Conversely, is it true that for every bipartite graph G there is a rational exponent r = r(G) such that t(n, G) = (nr )? (35) A Turn problem for even cycles a (Proposed by Erds [70]) o Prove that t(n, C2k ) cn1+1/k . A lower bound of order n1+1/(2k-1) can be proved by probabilistic methods [131]. The bipartite Ramanujan graph [180, 186] gives t(n, C2k ) n1+2/3k . Recently, Lazebnik, Ustimenko and Woldar [178] constructed graphs which u yield t(n, C2k ) n1+2/(3k-3) . F redi [146, 148] determined the exact values of t(n, C4 ) for infinitely many n. This conjecture is open except for the case of C4 , C6 and C10 (see Benson [21] and also Wenger [215] for a different construction). (36) A problem on Turn numbers for an n-cube a (proposed by Erds and Simonovits [130], 1970) o Let Qk denote an k-cube on 2k vertices. Determine t(n, Qk ). In particular, determine t(n, Q3 ). Erds and Simonovits [130] proved that o t(n, Q3 ) cn8/5 . No better lower bound than cn3/2 is known. (37) A problem on Turn numbers for graphs with degree constraints a (proposed by Erds and Simonovits [85], $250 for a proof and $100 for a o counterexample) Prove or disprove t(n, H) < cn3/2 if and only if H does not contain a subgraph each vertex of which has minimum degree > 2. (38) Turn numbers in an n-cube ($100, from the 70's, see [84]) a Let f2k (n) denote the maximum number of edges in a subgraph of Qn containing no C2k . Prove or disprove 1 f4 (n) = ( + o(1))n2n-1 . 2
9
It is known that 2k = limn f2k (n)/e(Qn ) exists [46]. The best bounds for 4 are .623 4 1/2 (see [46]). For larger k, it is known that 2 - 1 6 1/3 where the upper bound can be found in [46] and the construction for the lower bound is due to Conder [58]. Also, it was proved that 4k = 0 for k 2. Is it true that 6 = 1/3? Is it true that 10 = 0? (39) A problem on the octahedron graph (proposed by Erds, Hajnal, Ss and Szemerdi [113]) o o e Let G be a graph on n vertices which contains no K2,2,2 and whose largest independent set has o(n) vertices. Is it true that the number of edges of G is o(n2 )? Erds and Simonovits [127] determined the Turn number for the octaheo a dron graph K2,2,2 as well as other Platonic graphs [206, 203]. (40) A problem on the Turn number of C3 and C4 a (proposed by Erds and Simonovits [128]) o Let t(n, C3 , C4 ) denote the smallest integer m that every graph on n vertices and m edges must contain C3 or C4 as a subgraph. Is it true that 1 t(n, C3 , C4 ) = n3/2 + O(n) ? 2 2 Erds and Simonovits [128] proved that t(n, C4 , C5 ) = o Subgraph enumeration (41) Problem (proposed by Erds) o For a graph G, let #(H, G) denote the number of induced subgraphs of G isomorphic to a given graph H. Determine f (k, n) = min(#(Kk , G) + #(Kk , G))
G 1 n3/2 2 2
+ O(n).
where G ranges over all graphs on n vertices and G denotes the complement of G. An old conjecture of Erds stated that a random graph should achieve the o minimum which however was disproved by Thomason. In [210], he showed 5 1 that f (4, n) < 33 n , f (5, n) < 0.90621-(2) n , and in general, f (k, n) < 4 5 k 0.936 21-(2) n . Franek and Rdl [140] gave a different construction o which is simpler but gives a slightly larger constant. (42) Conjecture (proposed by Erds and Simonovits [129]) o Every graph G on n vertices and t(n, C4 ) + 1 edges contains at least two copies of C4 when n is large. Radamacher first observed (see [72]) that every graph on n vertices and t(n, K3 )+1 edges contains at least n/2 triangles. Similar question can be asked for a general graph H, but relatively few results are known for such problems (except for some trivial cases such as stars or disjoint edges). 10
k
(43) A conjecture on enumerating graphs with a forbidden subgraph (proposed by Erds, Kleitman and Rothschild [115]) o Denote by fn (H) the number of (labelled) graphs on n vertices which do not contain H as a subgraph. Then fn (H) < 2(1+o(1))t(n,H) . If H is not bipartite, this was proved by Erds, Frankl and Rdl [98]. o o For the bipartite case, it is open even for H = C4 . It is well known that t(n, C4 ) = (1/2 + o(1))n3/2 . On the other hand, Kleitman and Winston [165] proved 3/2 fn (C4 ) < 2cn . Recently, Kleitman and Wilson [164] proved that fn (C2k ) < 2cn for k = 3, 4, 5 and Kreuter [171] showed that the number of graphs on n 1+1/k vertices which do not contain C2j for j = 2, . . . , k is at most 2(ck +o(1))n where ck = .54k + 3/2. (44) A problem on regular induced subgraphs (Proposed by Erds, Fajtlowicz and Staton [82]) o Let f (n) be the largest integer for which every graph of n vertices contains a regular induced subgraph of f (n) vertices. Ramsey's theorem implies that a graph of n vertices contains a trivial subgraph, i.e., a complete or empty subgraph of c log n vertices. Conjecture: f (n)/ log n . Note that f (5) = 3 (since if a graph on 5 vertices contains no trivial subgraph of 3 vertices then it must be a pentagon). f (7) = 4 was proved by Fajtlowicz, McColgan, Reid, and Staton [137] and also by Erds and o Kohayakawa (unpublished). McKay (personal communication) found that f (16) = 5 and f (17) = 6. Bollobs observed that f (n) < n1/2+ for n a sufficiently large (unpublished). (45) A problem of Erds and McKay [82], 1994 ($100) o Let f (n, c) denote the largest integer m such that a graph G on n vertices containing no clique or independent set of size c log n must contain an induced subgraph with exactly i edges for each i, 0 < i m. Prove or disprove that f (n, c) n2 . McKay wrote, "It is easy to get bounds of the form f (n, c) c log n, and Paul had a more complicated way to prove a bound f (n) c (log n)2 , but I cannot remember it." Calkin, Frieze and McKay [42] proved that a random graph with pn2 edges, for a constant p, contains an induced subgraph with exactly i edges for each i, for i ranging from 0 up to (1 - )pn2 .
1+1/k
11
(46) A conjecture of Erds and Tuza [85] o Let G denote a graph on n vertices n2 /4 + 1 edges containing no K4 . Denote by f (n) the largest integer m for which there are m edges e in the complement of G so that G + e contains a K4 . Conjecture: n2 f (n) = (1 + o(1)) . 16 On triangle-free graphs (47) A problem on making a triangle-free graph bipartite (proposed by Erds, Faudree, Pach and Spencer [93]) o Is it true that every triangle-free graph on 5n vertices can be made bipartite by deleting at most n2 edges? This conjecture is proved for graphs with at least 5n2 edges [103]. Thus, the general conjecture is open for graphs with e edges for 2n2 < e 5n2 . (48) A problem on the number of triangle-free graphs (very recent, [205]) Determine or estimate the number of maximal triangle-free graphs on n vertices. (49) A problem on the number of pentagons in a triangle-free graph [72] Is it true that a triangle-free graph on 5n vertices can contain at most n5 pentagons? Gyri [152] proved that such graph can have at most 33 54 /214 n5 1.03n5 o triangles. (50) Conjecture (proposed by Erds, Faudree, Rousseau and Schelp [96]) o If each set of n/2 vertices in a graph of n vertices spans more than n2 /50 edges, then G contains a triangle. Krivelevich [172] proved that if each n/2 vertices span more than n2 /36 edges, then there is a triangle. (51) A problem on graphs covered by triangles (proposed by Erds and Rothschild [86]) o Suppose G is a graph of n vertices and e = cn2 edges. Assume that every edge of G is contained in at least one triangle. Determine the largest integer m = f (n, c) such that in every such graph there is an edge contained in at least m triangles. Alon and Trotter showed that f (n, c) < c n (personal communication). In the other direction, Szemerdi observed that the regularity lemma ime plies that f (n, c) approaches infinity for every fixed c. Is it true that f (n, c) > n ?
12
Graph coloring problems A graph is said to have chromatic number (G) = k if its vertices can be colored in k colors such that two adjacent vertices have different colors and such a coloring is not possible using k - 1 colors. (52) A problem on graphs with fixed chromatic number and large girth (proposed by Erds, 1962 [68]) o Let gk (n) denote the largest integer m such that there is a graph on n vertices with chromatic number k and girth m. Is it true that for k 4,
n
lim
gk (n) log n
exists? Erds [64, 68] proved that o log n 2 log n gk (n) + 1. 4 log k log(k - 2) (53) A problem on the chromatic number and clique number (proposed by Erds, 1967 [89]) o Let (G) denote the number of vertices in a largest complete subgraph of G. Let f (n) denote the maximum value of (G)/(G) where G ranges over all graphs on n vertices. Does the following limit exist? lim f (n) n/ log2 n
n
Erds [89] proved that o cn cn f (n) . 2 log n log2 n (54) A conjecture on subgraphs of given chromatic number and girth (proposed by Erds and Hajnal [75]) o For integers k and r, there is a function f (k, r) such that every graph with chromatic number at least f (k, r) contains a subgraph with chromatic number k and girth r. Rdl [192] proved the above conjecture for the case of r = 4 and for every o k. However, his upper bound is quite large. Erds [75] further conjectured: o lim f (k, r + 1) = . f (k, r)
k
(55) The problem of Erds and Lovsz [69] o a Suppose a graph G is k-chromatic and contains no Kk . Let a and b denote 13
two integers satisfying a, b 2 and a+b = k+1. Do there exist two disjoint subgraphs of G of chromatic numbers a and b, respectively? The original question of Erds is for the case k = 5, a = b = 3, which o was proved affirmatively by Brown and Jung [34]. Several small cases have been solved (for more discussion, see [158]). Of special interest is the following case for a = 2: Suppose the chromatic number of G decreases by 2 by removing any two vertices joining by an edge. Must G be the complete graph? (56) A problem on choosability of graphs (proposed by Erds, Rubin and Tayo lor [125]) A graph G is said to be (a, b)-choosable if for any assignment of a list of a colors to each of its vertices there is a subset of b colors of each list so that subsets corresponding to adjacent vertices are disjoint. Conjecture: If G is (a, b)-choosable, then G is (am, bm)-choosable for every positive integer m. The conjecture is known [10] to hold for graphs with n vertices, provided m is divisible by all integers smaller than some f (n). A special case of the above conjecture is the following problem (see [158]): Let G and H denote two graphs with the same set of vertices. If G is r-choosable (i.e., (r, 1)-choosable) and H is s-choosable, then their union is rs-choosable. (57) A problem on critical graphs (proposed by Erds in 1949 [76]) o A graph with chromatic number k is said to be edge critical or k-critical if the deletion of any edge decreases the chromatic number by 1. What is the largest number m, denoted by f (n, k), such that there is a k-critical graph on n vertices and m edges? In other words, determine lim f (n, k) = ck . n2
n
Edge critical graphs were first introduced by Dirac [62] who answered a problem of Erds from 1949 by showing f (n, k) > ck n2 for k 6 and, o o in particular, f (n, 6) > n2 /4 + cn. Erds and Simonovits proved that f (n, 4) < n2 /4 + cn. Toft [212] showed that f (n, 4) > n2 /16 + cn by using a graph with many vertices of bounded degree. Erds further raised the o following questions: (58) Is it true that f (n, 6) = n2 /4 + n for n 2 modulo 4? (59) Is there a 4-chromatic critical graph on n vertices and cn2 edges which does not contain Kt,t for some large t? Rdl (unpublished) constructed such an example with t < c log n. o 14
(60) A problem on critical graphs with large degree (proposed by Erds [76, 73]) o Let g(n, k) denote the maximum value m such that there exists a k-critical graph on n vertices with minimum degree at least m. What is the magnitude of g(n, k)? Is it true that g(n, 4) cn for some constant c? Simonovits [204] and Toft [213] proved that g(n, 4) cn1/3 . (61) A problem on vertex critical graphs (proposed by Erds [73]) o A graph with chromatic number k is said to be vertex critical or k-vertexcritical if the deletion of any vertex decreases the chromatic number by 1. Is there some positive function f (n) so that for every k 4 there exists a graph G on n vertices which is k-vertex-critical but (G - A) = k for any set A of at most f (n) edges? Brown [31] gave an example of a 5-vertex-critical graph with no critical edges. Recently, Jensen [157] showed that there exists a k-vertex-critical graph, for k 5, such that the chromatic number is not decreased after deleting any m edges all incident to a common vertex. (62) A conjecture on strong chromatic index (proposed by Erds and Neetil o s r [76] 1985) The strong chromatic index (G) of a graph G is the least number r so that the edges of G can be colored in r colors in such a way that any two adjacent vertices in G are not incident to edges of the same color. Suppose G has maximum degree k. Is it true that (G) 5k 2 /4 if k is even and (G) 5k 2 /4 - k/2 + 1/4 if k is odd? This conjecture is open for k 4 while the cases of k 3 are solved by Anderson [13] and Hork, Qing and Trotter [156]. Chung, Gyrfs, a a a Trotter and Tuza [56] proved that if G contains no induced 2K2 , then G has at most 5k 2 /4 edges. (63) A problem on three-coloring (proposed by Erds, Faudree, Rousseau and o Schelp [139]) Is it true that in every three-coloring of the edges of Kn there is a set of three vertices which are adjacent to at least two-third of all the vertices by edges of the same color? If true, it is the best possible as shown by an example given by Kierstead [139]. (64) A problem on anti-Ramsey graphs (proposed by Burr, Erds, Graham and Ss [39]) o o For a graph G, determine the least integer r = f (n, e, G) so that there is some graph H on n vertices and e edges which can be r-edge-colored such that all edges of every copy of G in H have different colors. 15
It seems to be a difficult problem to get good bounds for f (n, e, G) for a general graph G (see [39]). Even for special cases, there are large gaps between known bounds. For example, it was shown in [39, 38] that f (n, e, C5 ) cn n for e = (1/4 + )n2 and f (n, e, C5 ) = O(n2 / log n) for e = (1/2 - )n2 . Also, f (n, e, P4 ) > cn n for e = n2 and f (n, e, P4 ) n 2 for n = n / exp(c log n). Covering and packing (65) A conjecture on covering by C4 's (proposed by Erds and Faudree [88]) o Suppose a graph G has 4n vertices with minimum degree at least 2n. Then G has n vertex-disjoint C4 's. Alon and Yuster [11] proved that for a fixed bipartite graph H on h vertices, a graph G with n vertices, where h divides n, can be covered by vertex-disjoint copies of H if the minimum degree of G is at least (1/2+ )n for n sufficiently large. (66) A problem on clique covering and clique partition (proposed by Erds, Faudree and Ordman [92]) o The clique covering number cc(G) of G is the least number of cliques that covers the graph. The clique partition number cp(G) is the least number of cliques that partition the edge set of G. Here Gn denotes a graph on n vertices. Determine the largest value c such that cp(Gn ) > cn2 cc(Gn ) for an infinite family of graphs Gn . An example was given in [92] with c = 1/64. Is there a sequence of graphs Gn such that cp(Gn ) - cc(Gn ) = n2 /4 + O(n)? In [40] it was shown cp(Gn ) - cc(Gn ) = n2 /4 - n3/2 /2 + n/4 + O(1). (67) The ascending subgraph decomposition problem (proposed by Alavi, Boals, Erds, Chartrand and Oellermann [5]) o Suppose G is a graph with n(n + 1)/2 edges. Prove that G can be edgepartitioned into subgraphs Gi with i edges such that Gi is isomorphic to a subgraph of Gi+1 for i = 1, . . . , n - 1. A special case is the decomposition of star forests into stars (which is the so-called suitcase problem of partitioning integers 1, . . . , n into k parts with given sums a1 , . . . , ak for any ai n and ai = n(n + 1)/2). The suitcase problem was solved by Ma, Zhou and Zhou [185, 184]. 16
General extremal problems (68) A conjecture on trees 1962, (proposed by Erds and Ss) o o Every graph on n vertices having at least n(k-1)/2+1 edges must contain as a subgraph every tree of k + 1 vertices, for n k + 1. This conjecture, if true, is best possible. Some asymptotic approximations of this conjecture were given by Komls and Szemerdi (unpublished). o e Also, this conjecture is proved for some special families of trees such as caterpillars. Brandt and Dobson [30] have proved the conjecture for graphs with girth at least 5. (69) The (n/2-n/2-n/2) conjecture (proposed by Erds, F redi, Loebl and Ss [99]) o u o Let G be a graph with n vertices and suppose at least n/2 vertices have degree at least n/2. Then G contains any tree on at most n/2 vertices. Ajtai, Komls and Szemerdi [4] proved the following asymptotic version: o e If G has n vertices and at least (1 + )n/2 vertices have degree at least (1 + )n/2, then G contains any tree on at most n/2 vertices if n is large enough (depending on ). Komls and Ss conjectured [99]: o o Let G be a graph with n vertices and suppose at least n/2 vertices have degree at least k. Then G contains any tree with k vertices. (70) A conjecture of Erds and Gallai, 1959 [100] o Every connected graph on n vertices can be edge-partitioned into at most (n + 1)/2 paths. Lovsz [179] showed that every graph on n vertices can be edge-partitioned a into at most n/2 cycles and paths. Pyber [190] showed that every connected graph on n vertices can be covered by at most n/2 + O(n3/4 ) paths. (71) A problem on clique transversals (proposed by Erds, Gallai and Tuza [101]) o Estimate the cardinality, denoted by (G), of a smallest set that shares a vertex with every clique of G. Denote by R(n) the largest integer such that every triangle-free graph on n vertices contains an independent set of R(n) vertices. Is it true that (G) n - R(n) ? From the results on Ramsey numbers r(3, k), we know that c n log n < R(n) < c n log n. So far, the best known bound [101] is (G) n - 2n + c for a small constant c. (72) A problem on the diameter of a Kr -free graph (proposed by Erds, Pach, Pollack and Tuza [119]) o Let G denote a connected graph on n vertices with minimum degree .
17
Show that if G is K2r -free and is a multiple of (r - 1)(3r + 2), then the diameter of G, denoted by D(G), satisfies D(G) 2(r - 1)(3r + 2) n + O(1) (2r2 - 1)
when n approaches infinity. If G is K2r+1 -free and is a multiple of 3r -1, then 3r - 1 n + O(1) D(G) r when n approaches infinity. In [119], bounds for the diameters of triangle-free or Ck -free graphs with given minimum degree were derived. (73) Problem ($100, many years ago) Is there a sequence A of density 0 for which there is a constant c(A) so that for n > n0 (A), every graph on n vertices and c(A)n edges contains a cycle whose length is in A. Erds said [81], "I am almost certain that if A is the sequence of powers o of 2 then no such constant exists. What if A is the sequence of squares? I have no guess. Let f (n) be the smallest integer for which every graph on n vertices and f (n) edges contains a cycle of length 2k for some k. I think that f (n)/n but that f (n) < n(log n)c for some c > 0." Alon pointed out that f (n) cn log n using the fact [29] that graphs with n vertices and ck 1+1/k edges contain cycles of all even lengths between 2k and 2kn1/k (by taking k to be about log n/2). (74) Conjecture (proposed by Erds [74]) o For n 3, any graph with 2n+1 - n - 1 edges is the union of a bipartite 2 2 graph and a graph with maximum degree less than n. (75) A decomposition problem about odd cycles (proposed by Erds and Graham [102]) o It is known that a complete graph on 2n vertices can be edge-partitioned into n bipartite graphs (and this is not true for 2n + 1). Suppose a complete graph on 2n + 1 is decomposed into n subgraphs. Let f (n) denote the smallest integer m such that one of the subgraphs must contain an odd cycle of length less than or equal to m. Determine f (n). Is f (n) unbounded? (76) A problem on almost bipartite graphs (proposed by Erds [82]) o Suppose G has the property that for every m, every subgraph on m vertices contains an independent set of size m/2 - k. Let f (k) denote the smallest number such that G can be made bipartite by deleting f (k) vertices. Recently, Reed (unpublished) proved the existence of f (k) by using graph minors. It would be of interest to improve the estimates for f (k).
18
Erds, Hajnal and Szemerdi [114] proved that for every > 0 there o e is a graph of infinite chromatic number for which every subgraph of m vertices contains an independent set of size (1 - )m/2. Erds remarked o that perhaps (1 - )m/2 can be replaced by m/2 - f (m) where f (m) tends to infinity arbitrarily slowly. (77) A problem of Erds and Rado [122] o What is the least number k = k(n, m) so that for every directed graph on k vertices, either there is an independent set of size n or the graph includes a directed path of size m (not necessarily induced)? Erds and Rado [122] give an upper bound of [2m-1 (n-1)m +n-2]/(2n- o 3). Mitchell and Larson [176] give a recurrence relation and obtain a bound of n2 for m = 3 and, more generally, of nm-1 for m > 3.
Random graphs
(78) Problem (proposed by Erds, see [9]) o Let G denote a random graph on n vertices and cn edges. What is the largest r, denoted by r(c), for which the probability that the chromatic number is r is at least some constant strictly greater than 0 (and independent of n)? This is open except for r = 3 (see [9]). Luczak [182] gave an asymptotic estimate: c . r(c) = (1 + o(1)) 2 log c However, the exact values are not known. (79) Problem (proposed by Erds, see [9]) o How accurately can one estimate the chromatic number of a random graph (with edge probability 1/2)? Prove or disprove that the error is more (much more) than O(1). Shamir and Spencer have an O(n1/2 ) upper bound [201]. An old problem raised by Erds and Rnyi [123] is to determine the chroo e matic number of a random graph with given edge density. This problem has almost been completely resolved due to the work of Matula [186], Shamir and Spencer [201], and Bollobs [23]. The order of the chroa matic number for both sparse and dense random graphs have been determined asymptotically. For random graphs with edge density p satisfying p n-5/6- , where > 0, Shamir and Spencer showed that the chromatic number almost surely takes on at most five different values. Luczak later on [181] proved that for such sparse random graphs, the chromatic number is concentrated at two values. Recently, Alon and Krivelevich [8] 19
showed that for p n-1/2- , the chromatic number is concentrated at at most two values (and for some values of p, in a single value). However, the concentration of the limit function of for dense graphs is not as well understood yet. (80) A conjecture on a spanning cube in a random graph (proposed by Erds and Bollobs, see [88]) o a A random graph on n = 2d vertices with edge density 1/2 contains an d-cube. Alon and Fredi [7] showed that this conjecture is true if the random u graph has edge density > 1/2 for n large enough. (81) Problem (proposed by Erds and Bollobs, see [9]) o a In a random graph (with edge probability 1/2), find the best possible c such that every subgraph on n vertices will almost surely contain an independent set of size c log n (where c depends on ). (82) Problem (proposed by Erds and Spencer [9]) o Start with n vertices and add edges at random one by one. If we stop when every vertex is contained in a triangle, is there a set of vertex disjoint triangles covering every vertex (except for at most two vertices)? The above question can be posed for other configurations as well [9]. In particular, Bollobs and Frieze [26] showed that stopping as soon as there a is no isolated vertex, then there already almost surely is a perfect matching if the number of vertices is even. If we stop when every vertex has degree at least 2, Ajtai, Komls and Szemerdi [3] and Bollobs [24] proved that o e a there already almost surely is a Hamiltonian cycle. Alon and Yuster [12] and Ruci ski [196] examined the threshold function of a random graph on n n vertices for the existence of n/|V (H)| vertex-disjoint copies of a given graph H. (83) A problem on monotone graph properties (proposed by Erds, Suen and Winkler [133]) o A graph property P is said to be monotone if every subgraph of a graph with property P also has property P . Start with n vertices and add edges one by one at random, subject to the condition that property P continues to hold. Stop when no more edges can be added. How many edges can such a graph have? The cases when P denotes "triangle-free", "bipartite", or "disconnected" were considered by Erds, Suen and Winkler [133]. The property "maxo imum degree bounded by k" was examined by Ruciski and Wormald n [197]. Many interesting cases are still open, including "C4 -free", "Kr -free" (for r 4), "k-colorable" (for k 3), "planar" and "girth > k".
20
Hypergraphs
Ramsey theory for hypergraphs A t-graph has a vertex set V and an edge set E consisting of some prescribed set of t-subsets of V . For t-graphs Gi , i = 1, . . . , k, let rt (G1 , . . . , Gk ) denote the smallest integer m satisfying the property that if the edges of the complete t-graph on m vertices are colored in k-colors, then for some i, 1 i k, there is a t-subgraph isomorphic to Gi with all t-edges in the i-th color. We denote rt (n1 , . . . , nk ) = rt (Kn1 , . . . , Knk ). Clearly r2 (n1 , . . . , nk ) = r(n1 , . . . , nk ). (84) Conjecture ($500) Is there an absolute constant c > 0 such that log log r3 (n, n) cn? This is true if four colors are allowed [110]. If just three colors are allowed, there is some improvement due to Erds and Hajnal (unpublished). o r3 (n, n, n) > ecn
2
log2 n
.
(85) Generalized Ramsey problems ($500) (proposed by Erds and Hajnal [78]) o Denote by ft (n, u, v) the largest value of k such that any coloring of the t-tuples of a set of n elements in blue and red, then there are either k elements all of whose t-tuples are in blue or there are v red t-tuples of a set of u elements. Clearly, for v = k , the Ramsey function rt (k, k) = n t satisfies ft (n, k, k ) = k. t Conjecture ht (n, k, gt (k) + 1) (log n)c where n is sufficiently large and gt is defined by
t
gt (k) =
1
gt (ui ) + t ui i=1
and the u's are as nearly equal as possible; gt (k) = 0 for k < t; and gt (t) = 1. Turn problems a (86) Turn's conjecture for 3-graphs a For an r-uniform hypergraph (or r-graph, for short) H, we denote by tr (n, H) the smallest integer m such that every r-graph on n vertices with m + 1 edges must contain H as a subgraph. When H is a complete graph on k vertices, we write tr (n, k) = tr (n, H). 21
In memory of Turn, Erds offered $1000 for the following conjecture: a o Conjecture [214]
n
lim
t3 (n, 4)
n 3
=
5 . 9
There are many different extremal constructions [32, 169] which show t3 (n, 4) 5 n (1 + o(n-1 )). The best upper bound for t3 (n, 4)/ n is 3 9 3 (-1 + 21)/6 = .5971 . . . due to Giraud (unpublished, mentioned in [41]). An excellent survey on this problem can be found in F redi [149]. u (87) Conjecture [214]
n
lim
t3 (n, 5)
n 3
=
1 . 4
(88) A conjecture for triple systems (proposed by Brown, Erds and Ss [33, 78]) o o Let f (n, k, r) denote the least integer m such that every 3-graph on n vertices with more than m triples contains an induced subgraph on k vertices with at least r edges. Prove that f (n, k, k - 3) = o(n2 ). Ruzsa and Szemerdi [198] settled an earlier conjecture of Erds by showe o ing f (n, 6, 3) = o(n2 ). -systems A family of sets Ai , i = 1, 2, . . . , is called a strong -system if the intersections Ai Aj for i = j are all identical. In other words, Ai Aj =
t
At if i = j.
A strong -system of k sets is also called a k-star. The family is called a weak -system if we only require that the sizes |Ai Aj | are all the same for i = j. (89) A problem on unavoidable stars (proposed by Erds and Rado [120] in 1960) o For given integers n and k, determine the smallest integer m, denoted by f (n, k), for which every family of sets Ai , i = 1, . . . , m, with |Ai | = n for all i, contains a k-star. Erds and Rado [120, 121] proved that o 2n < f (n, 3) 2n n!. Abbott and Hanson [1] proved f (n, 3) > 10n/2 . Spencer [208] showed f (n) < (1 + )n n!.
22
(90) Conjecture ($1000): f (n, 3) cn for some absolute constant c. The current best bound is due to Kostochka [168]: f (n, 3) < n! (91) Conjecture : f (n, k) cn k (92) A problem on unavoidable stars of an n-set (proposed by Erds and Szemerdi [136]) o e Determine the least integer m, denoted by f (n, k) such that for any family A of subsets of an n-set with |A| > f (n, k), A must contain a k-star. Erds and Szemerdi [136] showed that f (n, 3) < 2(1-1/(10 n))n . Reo e cently, Deuber, Erd, Gunderson, Kostochka and Meyer [61] proved that o f (n, r) > 2n(1-log log r/2r-O(1/r)) for every r 3 and infinitely many n. In particular, f (n, 3) > 1.551n-2 for infinitely many n. (93) A problem on weak -systems (proposed by Erds, Milner and Rado [118]) o Let g(n, k) denote the least size for a family of n-sets forcing a weak system of k sets. Conjecture[118]: g(n, 3) < cn . k Recently, Axenovich, Fon-der-Flaass and Kostochka [14] proved g(n, 3) < (n!)1/2+ . (94) A problem on weak -systems of an n-set (proposed by Erds and Szeo merdi [136]) e Determine the least integer m, denoted by g (n, k), such that for any family A of subsets of an n-set with |A| > g (n, k), A must contain a weak -system of k sets. Erds and Szemerdi [136] proved that (n, g 3) > nlog n/4 log log n . Reo e 4/5 1 1/5 cently, Rdl and Thoma [194] proved that g (n, r) 2 3 n log (r-1) for o o r 3. For the upper bound for g (n, r) , Frankl and Rdl [142] proved that g (n, k) < (2 - )n , where depends only on k. (95) A problem of Erds, Faber and Lovsz ($500, 1972 [75]) o a Let G1 , . . . , Gn be n edge-disjoint complete graphs on n vertices. Then the chromatic number of n Gi is n. i=1 Recently, Kahn [159] proved that the chromatic number of n Gi is at i=1 most (1 + o(1))n. Erds also asked the question of determining n Gi if o i=1 we require that Gi Gj , i = j, is triangle-free, or should have at most one edge. 23
c log log n log n
n
.
(96) A problem on jumps in hypergraph ($500, proposed by Erds [75]) o Prove or disprove that any 3-uniform hypergraph with n > n0 vertices and at least (1/27 + )n3 edges contains a subgraph on m vertices and at least (1/27 + c)m3 edges where c > 0 does not depend on and m. Originally, Erds asked the question of determining such a jump for the o maximum density of subgraphs in hypergraphs with any given edge density. However, Frankl and Rdl [143] gave an example showing for hypero graphs with a certain edge density, there is no such jump for the density of subgraphs. Still, the original question for 3-uniform hypergraphs as described above remains open. (97) A problem on property B (proposed by Erds 1963, [66]) o A family F of subsets is said to have property B if there is a subset S such that every subset in F contains an element in S and an element not in S. What is the minimum number f (n) of subsets in a family F of n-sets not having property B? Property B is named after Felix Bernstein who first introduced this property in 1908 [22]. The best known upper bound is due to Erds [66, 67] o and the following lower bound was given by Beck [20]. n1/3- 2n f (n) < (1 + ) e log 2 2 n n 2 4
for n n0 and n0 depends only on . This problem was extensively considered in [9, 158]. (98) A conjecture on covering a hypergraph (proposed by Erds and Lovsz [117]) o a Let f (n) denote the smallest integer m such that for any n-element sets A1 , . . . , Am with Ai Aj = for i = j and for every set S with at most n - 1 elements, there is an Ai disjoint from S. Erds and Lovsz [117] o a proved that 8 n - 3 f (n) cn3/2 log n. 3 Kahn [160] showed that f (n) = O(n) and he wrote an excellent survey paper [161] on several related hypergraph problems (including this problem). Erds [84] further conjectured a strengthened version of this conjecture: o For every c > 0 there is an > 0 such that if n is sufficiently large and {Ai : 1 i cn} is a collection of intersecting n-sets, then there is a set S satisfying that |S| < n(1 - ) and Ai S = for all 1 i cn). (99) A problem on unavoidable hypergraphs (proposed by Chung and Erds [49]) o A r-graph H is said to be (n, e)-unavoidable if H is contained in every rgraph on n vertices and e edges. Let fr (n, e) denote the largest integer m 24
with the property that there exists an (n, e)-unavoidable r-graph having m edges. Determine fr (n, e). For the case of r = 2 and 3, the solutions can be found in [48, 49]. (100) A problem on unavoidable stars (proposed by Duke and Erds [91]) o Let f (n, r, k, t) denote the smallest integer m with the property that any r-graph on n vertices and m edges must contain a k-star with common intersection of size t. Determine f (n, r, k, t). Duke and Erds proved that f (n, r, k, 1) cnr-2 where c depends only o on r and k. For the case of r = 3, tight bounds are obtained by Chung and Frankl [54], also see [141, 47]. (101) A problem on decompositions of hypergraphs (proposed by Chung, Erds and Graham [51]) o For r-graphs H1 , . . . , Hk with the same number of edges, a U -decomposition (first suggested by Ulam) is a family of partitions of each of the edge sets E(Hi ) into t mutually isomorphic sets, i.e., say E(Hi ) = t Eij , where j=1 for each j, all the Eij are isomorphic. Let Uk (n, r) denote the least possible value m such that all families of k r-graphs must have a U -decompositions into t isomorphic sets. For graphs, it was shown [53, 50] that 1 2 2 n - < U2 (n, 2) < n + c 3 3 3 and for k 3, 3 3 n - n - 1 < Uk (n, 2) < n + ck . 4 4 There is still room for improvement.
For hypergraphs, it is of interest to determine U2 (n, 3), for example. It is known (see [51]) that c1 n4/3 log log n/ log n < U2 (n, 3) < c2 n4/3 . Also, for > 0, c3 n2-2/k- < Uk (n, 3) < c4 n2-1/k . (102) A problem on the product of the point and line covering numbers (proposed by Chung, Erds and Graham [52]) o In a hypergraph G with vertex set V and edge set E, the point covering number 0 (G) denotes the minimal cardinality of a subset of V which has non-empty intersection with every edge e in E. The line covering number 1 (G) denotes the minimal cardinality of a subset S of E such that every vertex is contained in some edge in S. The problem of interest 25
is to characterize hypergraphs which achieve the maximum and minimum value of 0 (G)1 (G). The case for graphs was solved in [52] and it was shown that n - 1 0 (G)1 (G) (n - 1) which is asymptotically best possible. Infinite graphs Erds wrote over a hundred papers on infinite graphs. In particular, the o problem papers by Erds and Hajnal [105, 106] contain 82 problems which have o been the major driving force in this field. Many of the problems have been solved positively, negatively or proved to be undecidable. The problems here are mainly based on the survey papers [86, 155, 153]. Many comments to these problems were graciously provided by Andrs Hajnal and Jean Larson. a Here we use the following arrow notation, first introduced by Rado: ( )r which means that for any r-partition f : []r there are < and H such that H has order type and f (Y ) = for all Y [H]r . If = for all < , then we write ()r . In this language, Ramsey's theorem can be written as ()r k for 1 r, k < . (103) A conjecture on ordinary partition relations for ordinals ($1000) (proposed by Erds and Hajnal [105]) o Determine the 's for which ( , 3)2 . Galvin and Larson [150] showed that such must be of the form . Chang [43] proved ( , 3)2 . Milner [187] generalized the proof of Chang to show ( , n)2 for n < , and Larson [173] gave a simpler proof. There have been many recent developments on ordinary partition relations for countable ordinals. Schipperus [199] proved that ( , 3)2
n+1 2
(2)
if is the sum of at most two indecomposables. In the other direction, Schipperus [199] and Larson [174] showed that ( , 5)2
(3)
26
if is the sum of two indecomposables. Darby [59] proved that ( , 4)2
(4)
if is the sum of three indecomposables. Schipperus [199] also proved that ( , 3)2 if is the sum of four indecomposables. (104) A problem on ordinary partition relations for ordinals (proposed by Erds and Hajnal [105]) o Is it true that if (, 3)2 , then (, 4)2 ? The original problem proposed in [105] was "Is it true that if (, 3)2 , then (, n)2 ?". However, Schipperus' results (2) and (3) give a negative answer for the case of n 5. For the case of n = 4, Darby and Larson (unpublished) proved ( , 4)2 extending the previous work of Darby on ( , 3)2 . (105) [105] Is it true that 1 (, 4)3 for < 1 ? Milner and Prikry [188] gave an affirmative answer for 2 + 1. (106) [105] Is it true that 1 2 (1 2 , 3)2 ? o A. Hajnal [154] proved 1 2 (1 2 , 3)2 under CH. Erds and Hajnal [106] ask if MA1 + 20 = 2 implies 1 2 (1 2 , 3)2 ? Erd, Hajnal and o Larson [108] asked for the cardinals that 2 (2 , 3)2 holds. Hajnal [154] showed the relation failed at successors of regular cardinals under GCH. Baumgartner [15] showed that the relation failed at successors of singular cardinals under GCH. (107) Is it true that 3 (2 + 2)3 ? Baumgartner, Hajnal and Todorevi [16] showed that GCH implies 3 c c (2 + )3 for < 1 and k < . k (108) A problem on graphs of infinite chromatic number ($250) (proposed by Erds, Hajnal and Szemerdi, 1982, [114, 82]) o e Let f (n) arbitrarily slowly. Is it true that there is a graph G of infinite chromatic number such that for every n, every subgraph of G of n vertices can be made bipartite by deleting at most f (n) edges? Prove or disprove the existence of a graph G of infinite chromatic number for which f (n) = o(n ) or f (n) = o((log n)c ). Rdl [191] solved this problem for 3-uniform hypergraphs. o 27
2 2 2 2
(5)
(109) A problem on 4-chromatic subgraphs (proposed by Erds, Hajnal [107]) o Is it true that if G1 , G2 are 1 -chromatic graphs then they have a common 4-chromatic subgraph? Erds, Hajnal and Shelah [112] proved that any 1 chromatic graph cono tains all cycles Ck for k > k0 . Consequently, the above problem has an affirmative answer for 3-chromatic graphs. (110) A problem on the union of triangle-free graphs ($250) (proposed by Erds and Hajnal [104]) o Is there a graph G which contains no K4 and which is not the union of 0 graphs which are triangle-free? Shelah [202] proved that the existence of such a graph is consistent but it is not known if this is provable in ZFC. (111) A problem on 1 -chromatic graphs (proposed by Erds, Hajnal and Szemerdi [114]) o e Is it true that if f (n) increases arbitrarily fast, then there is an 1 chromatic graph G so that if g(n) is the smallest integer for which G has an n-chromatic subgraph of g(n) vertices, then f (n)/g(n) 0? (112) A problem on odd cycles (proposed by Erds and Hajnal [87]) o Let G be a graph of infinite chromatic number and let n1 < n2 < . . . be the sequence consisting of lengths of odd cycles in G. Is it true that 1 = ? ni Gyrfs, Komls and Szemerdi [151] proved that the set of all cycle a a o e lengths has positive upper density. (113) A problem on ordinal graphs and infinite paths (proposed by Erds, Hajnal and Milner [109]) o For which limit ordinals is it true that if G is a graph whose vertices form a set of type then either G has an infinite path or contains an independent set of type . In other words, determine the limit for which (, infinite path)2 . Erds, Hajnal and Milner [109] proved that the positive relation is true o for all limit < +2 . Baumgartner and Larson [17] showed that if Jensen's Diamond Principle holds, then (, infinite path)2 for all +2 with 1 < 2 . Larson [175] obtained further results under the assumption of GCH. (114) A problem on ordinal graphs and down-up matchings (proposed by Erds and Larson [116]) o
28
A down-up matching in an ordinal graph is a matching of a set A with a set B where every element of A is less than all elements of B, denoted by A < B. Suppose for every graph on an ordinal there is either an independent set of type or a down-up matching from a A to a set B. If A has order type , then we write (, - matching)2 . Suppose that j and k are positive integers with k 2 and is a limit ordinal. Is it true that +jk ( +j , k - matching)2 ? If j and k 2 are positive integers and a countable limit ordinal, then Erds and Larson have shown that +jk+1 ( +j , - matching)2 but o +jk-1 ( +j , - matching)2 . Acknowledgement The author wishes to thank Paul Seymour for suggesting writing this paper. The author is especially grateful to Ron Graham who provided critical help in finding hundreds of references. During the preparation of this paper, many people have given numerous valuable comments and suggestions. Thanks are due to Noga Alon, Dan Archdeacon, Laci Babai, Jim Baumgartner, Jzsef o Beck, Gary Chartrand, Siemion Fajtlowicz, Zoltn Fredi, Jerry Grossman, a u Andrs Gyrfs, Andrs Hajnal, Jeong Han Kim, Yoshi Kohayakawa, Michael a a a a Krivelevich, Jnos Komls, Jean Larson, Tomasz Luczak, Brendan McKay, Jim a o Shearer, Miki Simonovits, Benny Sudakov, Lubos Thoma, Bjarne Toft, Zsolt Tuza, Vojta Rdl, Herb Wilf, Peter Winkler and Nick Wormald. o
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[51] F. R. K. Chung, P. Erds and R. L. Graham, Minimal decompositions of o hypergraphs into mutually isomorphic subhypergraphs, J. Comb. Theory Ser. A 32 (1982), 241251. [52] F. R. K. Chung, P. Erds and R. L. Graham, On the product of the point o and line covering numbers of a graph, Second International Conference on Combinatorial Mathematics (New York, 1978), Ann. N. Y. Acad. Sci. 319, 597602, New York Acad. Sci., New York, 1979. [53] F. R. K. Chung, P. Erds, R. L. Graham, S. M. Ulam and F. F. Yao, o Minimal decompositions of two graphs into pairwise isomorphic subgraphs, Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, FL, 1979), Congress. Numer. XXIII, 318, Utilitas Math., Winnipeg, Man., 1979. [54] F. R. K. Chung and P. Frankl, The maximum number of edges in a 3graph not containing a given star, Graphs and Combinatorics 3 (1987), 111126. [55] F. R. K. Chung and R. L. Graham, On multicolor Ramsey numbers for complete bipartite graphs, J. Comb. Theory Ser. B 18 (1975), 164169. [56] F. R. K. Chung, A. Gyrfs, W. T. Trotter and Zs. Tuza, The maximum a a number of edges in 2K2 -free graphs of bounded degree, Discrete Math. 81 (1990), 129135. [57] V. Chvtal, V. Rdl, E. Szemerdi and W. T. Trotter, The Ramsey numa o e ber of a graph with bounded maximum degree, J. Comb. Theory Ser. B 34 (1983), 239243. [58] M. Conder, Hexagon-free subgraphs of hypercubes, J. Graph Theory 17 (1993), 477479. [59] C. Darby, Negative partition relations for ordinals , (submitted to J. Comb. Theory, Ser A). [60] W. Deuber, Generalizations of Ramsey's theorem, Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erds on his 60th birthday), o Vol. I; Colloq. Math. Soc. Jnos Bolyai, Vol. 10, 323332, North-Holland, a Amsterdam, 1975. [61] , W. A. Deuber, P. Erds, D. S. Gunderson, A. V. Kostochka, A. G. o Meyer, Intersection Statements for Systems of Sets, J. Comb. Theory, Ser A, to appear. [62] G. A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952), 8592.
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[63] P. Erds, Extremal problems in number theory, combinatorics and geomo etry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 5170, PWN, Warsaw, 1984. [64] P. Erds, Graph theory and probability, Canad. J. Math. 11 (1959), 34 o 38. [65] List of publications of P. Erds, The Mathematics of Paul Erds, II (R. o o L. Graham and J. Neetil, eds.), 477-573, Springer-Verlag, Berlin, 1996. s r [66] P. Erds, On a combinatorial problem, Nordisk Mat. Tidskr. 11 (1963), o 510, 40. [67] P. Erds, On a combinatorial problem, II, Acta Math. Acad. Sci. Hungar. o 15 (1964), 445447. [68] P. Erds, On circuits and subgraphs of chromatic graphs, Mathematika 9 o (1962), 170175. [69] P. Erds, Problem 2, Theory of Graphs (Proc. Colloq., Tihany, 1966), o pp. 361, Academic Press, New York, 1968. [70] P. Erds, On sequences of integers no one of which divides the product of o two others and on some related problems, Mitt. Forsch.-Inst. Math. Mech. Univ. Tomsk 2 (1938), 7482. [71] P. Erds, On some of my favourite theorems, Combinatorics, Paul Erds o o is Eighty, Vol. 2, (D. Mikls, V. T. Ss, T. Sznyi, eds.), Bolyai Soc. Math. o o o Studies, 2 (1996), 97132. [72] P. Erds, On some problems in graph theory, combinatorial analysis and o combinatorial number theory, Graph Theory and Combinatorics (Cambridge, 1983), 117, Academic Press, London-New York, 1984. [73] P. Erds, Problems and results in combinatorial analysis and combinatoro ial number theory, Graph Theory, Combinatorics and Applications, Vol. 1 (Kalamazoo, MI, 1988), Wiley-Intersci. Publ., 397406, Wiley, New York, 1991. [74] P. Erds, Problems and results in graph theory, The theory and applicao tions of graphs (Kalamazoo, MI, 1980), 331341, Wiley, New York, 1981. [75] P. Erds, Problems and results in graph theory and combinatorial analysis, o Graph Theory and Related Topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977), 153163, Academic Press, New York-London, 1979. [76] P. Erds, Problems and results on chromatic numbers in finite and infinite o graphs, Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, MI, 1984), Wiley-Intersci. Publ., 201213, Wiley, New York, 1985. 34
[77] P. Erds, Problems and results on finite and infinite graphs, Recent o advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), 183192 (loose errata), Academia, Prague, 1975. [78] P. Erds, Problems and results on graphs and hypergraphs: similarities o and differences, Mathematics of Ramsey Theory, Algorithms Combin., 5, 1228, Springer, Berlin, 1990. [79] P. Erds, Some extremal problems on families of graphs and related probo lems, Combinatorial Mathematics (Proc. Internat. Conf. Combinatorial Theory, Australian Nat. Univ., Canberra, 1977), Lecture Notes in Math., 686, 1321, Springer, Berlin, 1978. [80] P. Erds, Some new problems and results in graph theory and other o branches of combinatorial mathematics, Combinatorics and graph theory (Calcutta, 1980), Lecture Notes in Math., 885, 917, Springer, Berlin-New York, 1981. [81] P. Erds, Some of my favorite problems and results, The Mathematics of o Paul Erds (R. L. Graham and J. Neetil, eds.), 4767, Springer-Verlag, o s r Berlin, 1996. [82] P. Erds, Some of my favourite problems in number theory, combinatorics, o and geometry, Combinatorics Week (Portuguese) (S~o Paulo, 1994), Rea senhas 2 (1995), 165186. [83] P. Erds, Some of my favourite theorems, Combinatorics, Paul Erds is o o Eighty, Volume 2, (D. Mikls, V.T. Ss and T. Sznyi, eds), Bolyai Society, o o o Mathematical Studies 2, Budapest 1993 and 1996. [84] P. Erds, Some of my favourite unsolved problems, A Tribute to Paul o Erds, 467478, Cambridge Univ. Press, Cambridge, 1990. o [85] P. Erds, Some of my old and new combinatorial problems, Paths, Flows, o and VLSI-layout (Bonn, 1988), Algorithms Combin., 9, 3545, Springer, Berlin, 1990. [86] P. Erds, Some problems on finite and infinite graphs, Logic and combinao torics (Arcata, Calif., 1985), Contemp. Math. 65, 223228, Amer. Math. Soc., Providence, R. I., 1987. [87] P. Erds, Some recent progress on extremal problems in graph theory, Proo ceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), Congress. Numer. XIV, 314, Utilitas Math., Winnipeg, Man., 1975. [88] P. Erds, Some recent combinatorial problems, Technical Report, Univero sity of Bielefeld, Nov. 1990.
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[89] P. Erds, Some remarks on chromatic graphs, Colloq. Math. 16 (1967), o 253256. [90] P. Erds, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. o 53 (1947), 292294. [91] P. Erds and R. Duke, Systems of finite sets having a common intero section, Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977), Congress. Numer. XIX, 247252, Utilitas Math., Winnipeg, Man., 1977. [92] P. Erds, R. Faudree and E. Ordman, Clique partitions and clique covo erings, Proceedings of the First Japan Conference on Graph Theory and Applications (Hakone, 1986), Discrete Math. 72 (1988), 93101. [93] P. Erds, R. Faudree, J. Pach and J. H. Spencer, How to make a graph o bipartite, J. Comb. Theory Ser. B 45 (1988), 8698. [94] P. Erds, R. Faudree, C. C. Rousseau and R. H. Schelp, A local density o condition for triangles, Graph Theory and Applications (Hakone, 1990), Discrete Math. 127 (1994), 153161. [95] P. Erds, R. J. Faudree, C. C. Rousseau and R. H. Schelp, On cycle o complete graph Ramsey numbers, J. Graph Theory 2 (1978), 5364. [96] P. Erds, R. Faudree, C. C. Rousseau and R. H. Schelp, The booktree o Ramsey numbers, Scientia, A: Mathematics 1 (1988), 111117. [97] P. Erds, R. J. Faudree, C. C. Rousseau and R. H. Schelp, The size Ramsey o number, Period. Math. Hungar. 9 (1978), 145161. [98] P. Erds, P. Frankl and V. Rdl, The asymptotic number of graphs not o o containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs and Combinatorics 2 (1986), 113121. [99] P. Erds, Z. F redi, M. Loebl and V. T. Ss, Discrepancy of trees, Combio u o natorics and its Applications to Regularity and Irregularity of Structures, (W.A. Deuber and V.T. Ss, eds), Studia Sci. Math. Hungar. 30 (1995), o 4757. [100] P. Erds and T. Gallai, On maximal paths and circuits of graphs, Acta o Math. Acad. Sci. Hungar. 10 (1959), 337356. [101] P. Erds, T. Gallai and Zs. Tuza, Covering the cliques of a graph with o vertices,Topological, Algebraical and Combinatorial Structures, Frol ik's memorial volume, Discrete Math. 108 (1992), 279289.
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