Introduction
These are the notes of the talk I gave on July 6th, 2011 in University of
Waterloo. The aim is to describe some results of the paper
Dependent Random Choice
by Jacob Fox and Benny Sudakov (Random Structures and Algorithms, 2011).
For some of the proofs, I have just used pages of the original paper.
Abbas Mehrabian
Warm-up
Lemma 1 (2.1). Assume that G has n vertices and average degree d, and there
is a t ∈ Z+ s.t.
dt
n
m t
−
≥ a,
r
nt−1
n
then ∃U ⊆ V of size ≥ a s.t. every r vertices in U have at least m common
neighbours.
Proof. Next page.
Define r(H) (Ramsey number of H) and Qr (the r-hyprecube graph). Burr
and Erdos conjectured r(Qr ) = O(2r ). It is known that
2
r(Qr ) ≤ 22r+o(r) < 22.7r < 23r < 8(16r)r < 2O(r ) .
Now we will see an easy proof for r(Qr ) < 23r , and later we will say a proof
for r(Qr ) < 22r+o(r) .
Theorem 2 (3.4).
r(Qr ) ≤ 23r .
Proof. Consider a 2-colouring of the complete graph on N = 23r vertices. Let
G be the graph of the denser colour, so its average degree is
1 N
N −1
2
×
=
≥ 2−4/3 N
d=
N
2 2
2
Let m = 2r , t = 3r/2. Then
dt
N
m t
−
≥
t−1
N
r
N
≥ 2r−1
So by previous lemma there is a subset U of size 2r−1 s.t. every r vertices in
U have at least 2r common neighbours. Embed one part of the hypercube
arbitrarily in U , and the other part can also be embedded.
Note: Lemma 1 cannot be extended to a and m both having linear size, even
if r = 2 and the graph has Θ(n2 ) edges.
70
FOX AND SUDAKOV
then G contains a subset U of at least a vertices such that every r vertices in U have at least
m common neighbors.
Proof. Pick a set T of t vertices of V uniformly at random with repetition. Set A = N(T ),
and let X denote the cardinality of A. By linearity of expectation,
t
|N(v)| t
dt
v∈V (G) |N(v)|
−t
t
1−t
E[X] =
=n
|N(v)| ≥ n
= t−1 ,
n
n
n
v∈V (G)
v∈V (G)
where the last inequality is by convexity of the function f (z) = zt .
Let Y denote the random variable counting the number of subsets S ⊂ A of size r with
fewer than m common neighbors. For a given such S, the probability that it is a subset of
t
. Since there are at most nr subsets S of size r for which |N(S)| < m, it
A equals |N(S)|
n
follows that
n
m t
E[Y ] <
.
r
n
By linearity of expectation,
dt
n
m t
E[X − Y ] ≥ t−1 −
≥ a.
n
n
r
Hence there exists a choice of T for which the corresponding set A = N(T ) satisfies
X − Y ≥ a. Delete one vertex from each subset S of A of size r with fewer than m common
neighbors. We let U be the remaining subset of A. The set U has at least X − Y ≥ a vertices
and all subsets of size r have at least m common neighbors.
3. A FEW QUICK APPLICATIONS
In this section we present four results which illustrate the application of the basic lemma to
various extremal problems.
3.1. Turán Numbers of Bipartite Graphs
For a graph H and positive integer n, the Turán number ex(n, H) denotes the maximum
number of edges of a graph with n vertices that does not contain H as a subgraph. A
fundamental problem in extremal graph theory is to determine or estimate ex(n, H). Turán
[85] in 1941 determined these numbers for complete graphs. Furthermore, the asymptotic
behavior of Turán numbers for graphs of chromatic number at least 3 is given by a well known
result of Erdős, Stone, and Simonovits (see, e.g., [9]). For bipartite graphs H, however, the
situation is considerably more complicated, and there are relatively few nontrivial bipartite
graphs H for which the order of magnitude of ex(n, H) is known. The following result
of Alon, Krivelevich, and Sudakov [3] is best possible for every fixed r, as shown by the
constructions in [55] and in [4]. Although, it can be derived also from an earlier result in
[46], the proof using dependent random choice is different and provides somewhat stronger
estimates.
Random Structures and Algorithms DOI 10.1002/rsa
Main Lemma
Let H be a bipartite graph with max degree ∆ and n vertices. It is known
that
2O(∆) n ≤ r(H) ≤ ∆2∆+3 n < 2∆ log ∆+O(∆) n < f (∆)n.
In particular, r(Qr ) ≤ r22r+3 .
The following theorem implies r(H) ≤ ∆2∆+3 n.
Theorem 3 (6.1, Fox and Sudakov’09). Assume > 0 and G has N ≥ 8∆−∆ n
vertices and ≥ N2 edges. Then G contains H.
Proof. Follows from the following two lemmas.
Lemma 4 (6.3). Assume G has N > 4d−d n vertices and at least N 2 /2 edges.
Then ∃U ⊆ V of size > 2n s.t. the fraction of d-sets S ⊆ U with |N (S)| < n is
less than (2d)−d .
Proof. Next page.
6.1
Dependent random choice lemma
A d-set is a set of size d. The following extension of Lemma 5.1 shows that every dense graph contains
a large set U of vertices such that almost every d-set in U has many common neighbors.
Lemma 6.3 If ϵ > 0, d ≤ n are positive integers, and G = (V, E) is a graph with N > 4dϵ−d n vertices
and at least ϵN 2 /2 edges, then there is a vertex subset U with |U | > 2n such that the fraction of d-sets
S ⊂ U with |N (S)| < n is less than (2d)−d .
Proof. Let T be a subset of d random vertices, chosen uniformly with repetitions. Set U = N (T ),
and let X denote the cardinality of U . By linearity of expectation and by convexity of f (z) = z d ,
E[X] =
∑ ( |N (v)| )d
v∈V
N
=N
−d
∑
(∑
|N (v)| ≥ N
d
1−d
v∈V
|N (v)|
N
)d
v∈V
≥ ϵd N.
Let Y denote the random variable counting the number of d-sets in U with fewer than n common
(
)d
neighbors. For a given d-set S, the probability that S is a subset of U is |NN(S)| . Therefore, we
have
)
( )(
N
n−1 d
.
E[Y ] ≤
N
d
By convexity, E[X d ] ≥ E[X]d . Thus, using linearity of expectation, we obtain
[
]
E[X]d
E[X]d
d
E X −
Y −
≥ 0.
2E[Y ]
2
Therefore, there is a choice of T for which this expression is nonnegative. Then
1
1 2
X d ≥ E[X]d ≥ ϵd N d
2
2
and hence |U | = X ≥ ϵd N/2 > 2n. Also,
) ( )
( )
( )( )
(
N
n d 1
2n d |U |
−d |U |
Y ≤ 2X E[Y ]E[X] < 2|U |
≤ (2d)
,
<
d
N
d
d
ϵd N
ϵd2 N d
( )
where we use that |U |d ≤ 2d−1 d! |Ud | which follows from |U | > 2n ≥ 2d.
d
6.2
−d
d
2
Embedding lemma and the proof of Theorem 6.1
Next we show how to embed a sparse bipartite graph in a graph containing a large vertex set almost
all of whose small subsets have many common neighbors. This will be used to deduce Theorem 6.1.
Lemma 6.4 Let H be a bipartite graph on n vertices with maximum degree d. If a graph G contains
a subset U such that |U | > 2n and the fraction of d-sets in U with less than n common neighbors is
less than (2d)−d , then G contains a copy of H.
12
Ramsey Number of Bipartite Graphs
Lemma 5 (6.4). If Lemma 4 is true with d = ∆ then G contains H.
Proof. Say a d-set is bad if its elements have less than n common neighbours, and
set S of size ≤ d is bad if a random extension of it is bad with prob ≥ (2d)|S|−d .
Claim. If S is good, then the number of x ∈
/ S with S ∪ {x} bad is ≤ |U |/2d.
Otherwise, the prob that a random extension of S is bad would be at least
|BAD(S)|
(2d)|S|+1−d > (2d)|S|−d
|U | − |S|
Let V (H) = V1 ∪ V2 and Li = {v1 , . . . , vi }.
Claim. There is an embedding f : V1 → U s.t. for each 1 ≤ i ≤ |V1 | and
w ∈ V2 , f (N (w) ∩ Li ) is good.
Proof. The empty set is good. Use induction on i, assume you want to define
f (vi+1 ). For every neighbour w of vi+1 , the set N (w) ∩ Li is good so there are
≤ |U |/2d vertices that are bad for it. This gives a total of ≤ |U |/2 vertices.
Since |U |/2 − i > 0 there is a vertex that is not bad for any of these sets, and
that is f (vi+1 ).
Then we embed vertices in V2 one by one. For each w ∈ V2 , the set f (N (w))
is good, so its elements have n common neighbours, embed w in an unoccupied
one.
Subdivisions of Complete Graphs
Theorem 6 (8.1, Alon, Krivelevich and Sudakov’03). If G has n vertices and
√
n2 edges, then G contains the 1-subdivision of the complete graph on n vertices.
Proof. First we prove a claim, whose proof is in the next page.
√
Claim (Lemma 8.2). Let k = n. Then ∃U ⊆ V of size k s.t. for all
1 ≤ i ≤ k2 , there are less than i pairs of vertices in U with fewer than i
common neighbours in G − U .
Now we prove the theorem.
Let S1 , . . . , S(k) be the set of pairs in U arranged such that
2
|N (S1 ) \ U | ≤ |N (S2 ) \ U | ≤ · · · ≤ |N (Si ) \ U | ≤ · · · ≤ |N (S(k) ) \ U |
2
By claim, |N (Si ) \ U | ≥ i. Let v1 ∈ N (S1 ) \ U , v2 ∈ N (S2 ) \ (U ∪ {v1 }) and so
on, in general
vi ∈ N (Si ) \ (U ∪ {v1 , . . . , vi−1 })
One can find all vi ’s. They build a 1-subdivision of Kk in G.
86
FOX AND SUDAKOV
less than i pairs of vertices in U with fewer than i common neighbors outside U. Indeed,
suppose we have found, in the graph G, a vertex subset U with |U| = k such that for each i,
1 ≤ i ≤ 2k , there are less than i pairs of vertices in U with fewer than i common neighbors
in G \ U. Label all the pairs S1 , . . . , S(k ) of vertices of U in non-decreasing order of the size
2
of |N(Si ) \ U|. Note that for all i we have by our assumption that |N(Si ) \ U| ≥ i. We find
distinct vertices v1 , . . . , v(k ) such that vi ∈ N(Si ) \ U. These vertices together with U form
2
a copy of the 1-subdivision of the complete graph of order k in G, where U corresponds
to the vertices of the complete graph, and each pair Si is connected by a path of length 2
through vi . We construct the sequence v1 , . . . , v(k ) of vertices one by one. Suppose we have
2
found v1 , . . . , vi−1 , we can let vi be any vertex in N(Si ) \ U other than v1 , . . . , vi−1 . Such a
vertex vi exists since |N(Si ) \ U| ≥ i. Thus to finish the proof of Theorem 8.1 we only need
to prove the following lemma.
1/2
Lemma 8.2. Let G = (V , E) be a graph with n vertices and n2 edges, and let
k k = n .
Then G contains a subset U ⊂ V with |U| = k such that for each i, 1 ≤ i ≤ 2 , there are
less than i pairs of vertices in U with fewer than i common neighbors in G \ U.
Proof. Partition V = V1 ∪V2 such that |V1 | = |V2 | = n/2 such that at least half of the edges
of G consisting of those edges that
of G cross V1 and V2 . Let G1 be the bipartite subgraph
cross V1 and V2 . Without loss of generality, assume that v∈V1 |NG1 (v)|2 ≤ v∈V2 |NG1 (v)|2 .
Pick a pair T of vertices of V1 uniformly at random with repetition. Set A = NG1 (T ) ⊆ V2 ,
and let X denote the cardinality of A. By linearity of expectation,
2
|NG (v)| 2
|N
(v)|
G
v∈V
1
1
2
E[X] =
= 4n−2
|NG1 (v)|2 ≥ 2n−1
≥ 2 2 n,
n/2
n/2
v∈V
v∈V
2
2
where the first inequality is by convexity of the function f (z) = z2 .
Define the weight w(S) of a subset S ⊂ V2 by w(S) = |N 1 (S)| . Let Y be the random
G1
variable which sums the weight of all pairs S of vertices in A. We have
|NG1 (S)| 2
w(S)P(S ⊂ A) =
w(S)
= 4n−2
|NG1 (S)|
E[Y ] =
n/2
S⊂V2 ,|S|=2
S⊂V2 ,|S|=2
S⊂V2 ,|S|=2
|NG (v)|
1
= 4n−2
< 2n−2
|NG1 (v)|2 ≤ 2n−2
|NG1 (v)|2 = E[X]/2
2
v∈V
v∈V
v∈V
1
1
2
This inequality with linearity of expectation implies E[X − E[X]/2 − Y ] > 0. Hence, there
is a choice of T such that the corresponding set A satisfies X > E[X]/2 ≥ 2 n and X > Y .
Let U be a random subset of A of size exactly k and Y1 be the random variable which
sums the weight of all pairs S of vertices in U. We have
k E[Y1 ] = X2 Y < (k/X)2 X = k 2 /X < 1.
2
This implies that there is a particular
subset U of size k such that Y1 < 1. In the bipartite
graph G1 , for each i, 1 ≤ i ≤ 2k , there are less than i pairs of vertices in U with fewer than
i common neighbors. Indeed, otherwise Y1 = S⊂U,|S|=2 |N 1 (S)| is at least i 1i = 1. Hence,
G1
in G, there are less than i pairs of vertices in U with fewer than i common neighbors in
G \ U.
Random Structures and Algorithms DOI 10.1002/rsa