MATH 7018: Homework 1 (due Jan 30 at 11:59pm)
Please typeset your homework solutions using L
A
T
EX.
We use the notation N:“ t0,1, . . .uand N`:“ t1,2, . . .u. For nPN`, we let rns:“ t1,2, . . . , nu.
Problem 1.
Fix integers
n
,
k
,
tPN`
. Show that if
tą
2
k
4
klog n
, then there are subsets
S1
, ...,
StĎ rns
such that for every pair of disjoint k-element subsets A,BĎ rns, there is some iP rtswith
AĎSiĎBc.
(Here Bc:“ rnszBdenotes the complement of B.)
Problem 2.
For the purposes of this problem, a permutation of length
n
is a sequence
σ“ pσ1, . . . , σnq
in which every element of the set
rns
appears exactly once. For a permutation
σ
, let
Lpσq
be
the maximum length of an increasing subsequence of
σ
. For instance,
Lp
1
,
5
,
2
,
4
,
3
,
6
q “
4, as
witnessed by the subsequence
p
1
,
2
,
4
,
6
q
. Suppose that a permutation
σ
of length
n
is chosen
uniformly at random. Prove that PrLpσq ě 100?ns “ op1q.
(Here op1qindicates a function fpnqsuch that limnÑ8 fpnq “ 0.)
Problem 3.
Let
H
be a 3-uniform hypergraph with
n
vertices and
m
edges. Recall that a subset
IĎVpHq
is independent if it contains no edge of
H
. We use
αpHq
to denote the maximum size of an
independent set in H. Show that if měn{3, then
αpHq ě 2
3?3¨n3{2
?m.
Problem 4.
Let
s
,
tPN`
be positive integers. Recall that the (off-diagonal)Ramsey number
Rps, tq
is
the smallest
N
such that every graph
G
on at least
N
vertices satisfies
ωpGq ě s
or
αpGq ě t
.
(a)Show that for any nPNand 0ďpď1,
Rps, tq ą n´ˆn
s˙pps
2q´ˆn
t˙p1´pqpt
2q.
(b)Apply part (a)with p“n´2{sto deduce that for all s,tě3,
Rps, tq ě csˆt
log t˙s{2
,
where csą0may depend on sbut not on t.
Problem 5.
Recall that a function
f
defined on a vertex set of a graph
G
is called a proper coloring of
G
if
fpuq ‰ fpvq
for every edge
uv PEpGq
. Let
G
be a bipartite graph with
n
vertices. Suppose
that for each vertex
vPVpGq
, we are given a set
Lpvq
of size
|Lpvq| ą log2n
. Prove that
G
has a proper coloring fwith fpvq P Lpvqfor all vPVpGq.
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