Solutions to Homework #1:
1) Let k < n be positive. Prove that (n/k )k <
Sol: (Lower bound)
n
k
n
k
< (ne/k )k .
n(n 1) (n k + 1)
=
=
k!
k1
i=0
ni
.
ki
It therefore suces to show that (n i)/(k i) > n/k for i 1, which is equivalent to
n > k.
(Upper bound)

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
that
Rk (3) 1 1 + k (Rk1 (3) 1).
Use this to prove the up

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
2
monochromatic (in ) complete graph with vertex set a1 < a2 < .

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

Solutions to Homework #5:
[n]
1) 9.1: Suppose that F k is an antichain. Show that |F |
Sol: The LYM inequality yields
|F |
n
k
1
A F
n
|A|
n
k
.
1
1.
2) 9.2:
a
Sol: Let Ai = Ai (Ai Bi ). Now apply Bollobs theorem to the set pairs (Ai , Bi ). Since
t
|Ai

Homework #6 Solutions:
1) 11.1
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,