1) Prove that in every 2-coloring of the edges of the complete graph on n
vertices (n 3), there is a hamiltonian cycle which is either monochromatic,
or consists of two monochromatic arcs.
2) Prove or disprove:
a) In every 3-coloring of the poi
Solutions to HW 6
1. Fix a point x on the circle. Moving clockwise, write 1 if a chord is encountered for the
rst time, and 1 otherwise. We obtain a sequence of 1s of length 2n satisfying the
Catalan properties. On the other hand, given such a
Solutions to HW 5
1. (a) It equals f2n . It is true for n = 1. Now the induction step:
f1 + f3 + + f2n+1 = (f1 + f3 + + f2n1 ) + f2n+1 = f2n + f2n+1 = f2n+2 = f2(n+1) .
(b) It equals f2n+1 1. It is true for n = 0. Now the induction step:
f0 + f
Solutions to HW 4
2. Let (4, 6, 7, 8) = (a1 , a2 , a3 , a4 ), and let Ai be the number of positive integers less than
10,000 divisible by ai . Then we seek N as dened in class. Clearly |Ai | = 10000/ai ,
|Ai Aj | = 10000/lcm(ai aj ) etc. Care
Solutions to HW 1
1. If both m and n are odd, then the number of squares to be covered is mn which is odd.
Each domino covers 2 squares, so the total number of squares covered by the dominoes
must be even. Hence a perfect cover is impossible. On the other
Solutions to HW 3
5. We proceed by induction on k (not n!), proving that if the sum is at least k , then we
need at least k switches. If k = 0, then the conclusion is trivial, so assume k > 0. Consider
the switch ij , ij +1 . Then bl remains
Solutions to HW 2
1. With no restriction, we have ve choices for each position, giving 54 numbers. If a)
holds, then there are ve choices for the rst digit, then four for the next and so on, giving
5 4 3 2 = 120. If b) holds, then the rst three
Homework #6 Solutions:
Sol: Since A is (n, k )-dense, there is a vector in A with at least k 1s, so r k . Let v be
a vector of A with the maximum number of 1s, say s. Then k s r. Now pick k 1s
from v . Since the corresponding k -set is shattered,
Solutions to Homework #5:
1) 9.1: Suppose that F k is an antichain. Show that |F |
Sol: The LYM inequality yields
Sol: Let Ai = Ai (Ai Bi ). Now apply Bollobs theorem to the set pairs (Ai , Bi ). Since
1) Consider the set S of all ordered k -tuples A = (A1 , . . . , Ak ) of subsets of
|A1 . . . Ak |.
2) Give an explicit bijection between the set of plane rooted trees with n
vertices and the set of plane rooted binary trees w
Homework #4 Solutions:
0) Continuation of 29.5 from previous homework. Show that
n + lg n 2 (f ) n + 2 lg n .
Sol: The problem is equivalent to showing that lg n (G) 2 lg n , where G is the
graph with vertex set V = ([n] [n]) cfw_(i, i) : i [n] and vertex
1) Problem 10E: You may assume that fn (z ) = (z ) where the product
is over all such that n = 1 and k = 1 for 1 k < n.
2) Suppose that f is a function from N to N such that f (a + b) f (a)+ f (b).
Prove that limn f (n)/n exists (possibly ).
Homework #3 Solutions:
27.1) Let n = Rr (2, l) as in the hint. Given an r-coloring of [n], dene an r-coloring
of [n] by (ij ) = (j i) for i < j . Then the choice of n implies that we get a
monochromatic (in ) complete graph with vertex set a1 < a2 < .
1) Let [S, T ] and [S , T ] be source/sink cuts in a network.
a) Prove that c(S S , T T ) + c(S S , T T ) c(S, T ) + c(S , T ).
b) Suppose that [S, T ] and [S , T ] are minimum cuts. Show that no edge
between S S and S S has positive capacity.
Homework #2 Solutions:
1) Dene the multicolor Ramsey number Rk (3) to be the minimum n such that no matter
how the edges of Kn are colored with k colors, there is a monochromatic copy of K3 . Prove
Rk (3) 1 1 + k (Rk1 (3) 1).
Use this to prove the up
1) Prove that a graph with n vertices and e edges has at least
2) Fix integers 2 s t. Prove that there is a constant c = c(s, t) such that
ex(n, Ks,t ) < cn21/t . Hint: use the argument from class for s = t = 2.
Solutions to Homework #1:
1) Let k < n be positive. Prove that (n/k )k <
Sol: (Lower bound)
< (ne/k )k .
n(n 1) (n k + 1)
It therefore suces to show that (n i)/(k i) > n/k for i 1, which is equivalent to
n > k.
Solutions to HW 7
42. For each r = 1, 2, 3, 4, let Lr be dened by Lr = ri + j for each i, j cfw_0, . . . , 5,
where everything is taken modulo 5. Then we proved in class that these are MOLS.
Chapter 14 :
7. Label the corners 1, 2, 3, 4. Th