PROBABILISTIC METHODS IN COMBINATORICS 1. Introduction ...

PROBABILISTIC METHODS IN COMBINATORICS
YUFEI ZHAO
Abstract. This is a talk I gave for the Cambridge Part III seminar at the end of Lent term 2011.
The probabilistic method, originally popularized by Paul Erdős, is a powerful technique in
combinatorics. In this talk, we provide an introduction to the probabilistic method using examples
from Ramsey theory and set systems.
1. Introduction
The probabilistic method is a tool for tackling combinatorics problem by introducing randomness.
For instance, to prove that a combinatorial object with certain property exists, we could construct
some object randomly, and then show that it has the desired property with positive probability.
Here is a simple example illustrating this idea.
Theorem 1.1. Every graph G = (V, E) contains a bipartite subgraph with at least |E|
2
edges.
Proof. Randomly assign each vertex of G with black or white, independently with uniform probability.
Consider the set of edges E ′ with different colours on its endpoints. Then (V, E ′ ) is a
bipartite subgraph of G. Note that every edge belongs to E ′ with probability 1 2
, so by linearity of
expectation, E[|E ′ |] = 1 2 |E|. Thus there is some colouring for which |E′ | ≥ 1 2
|E|, and this gives
the desired bipartite subgraph.
□
2. Ramsey numbers
The Ramsey number R(k, l) is defined to be the smallest n such that if the edges of the complete
graph K n are coloured red or blue, then it contains either a copy of a red K k or a copy of a blue
K l . It was shown by Ramsey that every R(k, l) is finite. However, very few exact values of R(k, l).
It is an active problem in combinatorics to study the growth behavior of these Ramsey numbers.
In this section, we give several lower bounds to the diagonal Ramsey numbers R(k, k), and with
each bound we introduce a new idea in the probabilistic method.
2.1. Lower bound to Ramsey numbers. The following lower bound to Ramsey numbers is
given by Erdő in a 1947 paper that “started” the probabilistic method.
Theorem 2.1. If ( n
k)
2
1−( k 2) < 1, then R(k, k) > n.
By optimizing n, this theorem gives us
R(k, k) > 1
e √ (1 + o(1))k2k/2
2
Proof. We need to show that there exists a colouring of the edges of K n with two colours containing
no monochromatic K k . Let us colour the edges of K n randomly. The probability that a particular
subgraph K k is monochromatic is exactly 2
2 (k 2) = (
2) 21−(k . By consider all n
)
k copies of Kk in K n , we
find that the probability that there is some monochromatic K k is at most ( n
k)
2
1−( k 2) < 1. Therefore,
with positive probability, the colouring has no monochromatic K k , thereby proving the existence
of such a colouring.
□
Date: 17 April 2011.
1

2 YUFEI ZHAO
2.2. Alterations. Next we give a slightly better lower bound to R(k, k) using the idea of alterations.
Our approach in the previous proof is that to randomly pick an edge-colouring of K n and
then hope that it contains no monochromatic K k . Alternatively, we can first pick a random edgecolouring
of K k , and then modify the graph to get rid of the bad parts, namely the monochromatic
K k .
( n
Theorem 2.2. For any k, n, we have R(k, k) > n − 2
k)
1−(k 2) .
By optimizing the choice of n, this theorem gives us
R(k, k) > 1 e (1 + o(1))k2k/2 ,
which improves the previous bound by a constant factor of √ 2.
Proof. Randomly colour the edges of K n with two colours. Whenever we see a monochromatic
K k , delete one of its vertices. As in the previous proof, the probability that a particular subgraph
K k is monochromatic is exactly 2 1−(k 2) , so the expected number of monochromatic Kk ’s is exactly
( n
)
k 2
1−( k 2) . Since we delete at most one vertex per every monochromatic Kk , we remove at most
( n
)
k 2
1−( k 2) vertices on expectation. Hence with some positive probability, the remaining graph has
at least n − ( n
k)
2
1−( k 2) vertices, and it has no monochromatic Kk . This proves the desired lower
bound to R(k, k).
□
2.3. Lovász local lemma. We give one more improvement to the lower bound, using the Lovász
local lemma, which we state without proof.
Theorem 2.3 (Lovász local lemma). Let E 1 , . . . , E n be events, with Pr[E i ] ≤ p for all i. Suppose
that each E i is mutually independent of all other E j except for at most d of them. If
ep(d + 1) < 1,
then with some positive probability, none of the events E i occur.
Here is some intuition about the local lemma. We can view the events E i as “bad events” that we
want to avoid. If they are all independent, then we know easily there is some positive probability
that none of the bad events occur as long as each bad event has probability less than 1. On the
other hand, if the probability of each E i is very small, say smaller than 1 n
, then we can apply the
union bound to see that there is some probability that none of them occur. The situation reflected
inn the local lemma is between these two extremes. We know that p is small, but not as small as
1
n
. We know that most of the events mutually independent, but not all are. The local lemma tells
us that even in this situation, we can conclude that there is some positive probability that none of
the bad events occur.
( (k )(
Theorem 2.4. If e n
) )
2 + 1 2 1−(n k) < 1, then R(k, k) > n.
k−2
By optimizing the choice of n, this theorem gives us
√
2
R(k, k) >
e (1 + o(1))k2k/2 ,
once again improving the previous bound by a constant factor of √ 2. This bound was given by
Spencer in 1975. It is the best known lower bound to R(k, k) to date.
Proof. Consider a random colouring of the edges of K n . For each subset R of the vertices of K n with
k vertices, let E R denote teh event that R induces a monochromatic K k . Then Pr[E R ] = 2 1−(k 2) .

PROBABILISTIC METHODS IN COMBINATORICS 3
The events E R and E S are independent when the subgraphs induced by R and S do not share
edges. Thus is E R and E S are dependent, then we necessarily have |R ∩ S| ≥ 2. For a fixed R,
there are at most ( )(
k n
2 k−2)
choices of S with |S| = k and |R ∩ S| ≥ 2.
Applying the Lovász local lemma to the events {E R : R ⊂ V (K n ), |R| = k} and p = 2 1−(k 2) and
d = ( )(
k n
2 k−2)
, we see that with positive probability none of the events ER occur, thereby giving a
colouring with no monochromatic K k ’s.
□
3. Set systems
In this section we apply the probabilistic method to two extremal problems regarding families of
subsets of {1, 2, . . . , n}.
3.1. Antichains. Let F be a collection of subsets of {1, 2, . . . , n} such that no set in F is contained
in another set in F. Such a collection is called an antichain. We would like to know what is the
maximum number of sets in an antichain.
If F is the collection of all size k subsets of {1, 2, . . . , n}, then F is an antichain, since no k-
element set can contain another k-element set. This gives |F| = ( n
k)
, which is maximized when
k = ⌊ n
2
⌋
or
⌈ n
2
⌉
. The next result shows that we cannot do better.
Theorem 3.1 (Sperner). If F is an antichain of subsets of {1, 2, . . . , n}, then |F| ≤
( ) n
.
⌊n/2⌋
Proof. Consider a random permutation σ of {1, 2, . . . , n}, and its associated chain of subsets
∅, {σ(1)} , {σ(1), σ(2)} , {σ(1), σ(2), σ(3)} , . . . , {σ(1), . . . , σ(n)}
where the last set is always equal to {1, 2, . . . , n}. For each A ⊂ {1, 2, . . . , n}, let E A denote the
event that A is found in this chain. Then
|A|!(n − |A|)!
Pr(E A ) = = 1 1
(
n!
n
) ≥ ( n
).
|A| ⌊n/2⌋
Indeed, if A were to be found in the chain, it must occur in the (|A| + 1)-th position, and are |A|!
ways to form a chain in front of it, and (n − |A|)! ways to continue the chain after it.
Since F is an antichain, if A, B ∈ F are distinct, then E A and E B cannot both occur. Thus
1
{E A : A ∈ F} is a set of disjoint events, each with probability at least
( ⌊n/2⌋) . Thus |F| ≤ ( n
⌊n/2⌋)
.
n
3.2. Intersecting family. We say that family F of sets is intersecting if A ∩ B ≠ ∅ for every
A, B ∈ F. Let F be an intersecting family of k-element subsets of {1, 2, . . . , n}. How large can |F|
be?
If we take F to be the collection of all k-element subsets of {1, 2, . . . , n} containing the element
1, then F is intersecting, and |F| ≤ ( n−1
k−1)
. Now we show that this is the best we can do.
Theorem 3.2 ( (Erdős-Ko-Rado). )
If F is a intersecting family of k-element subsets of {1, 2, . . . , n},
n − 1
then |F| ≤ .
k − 1
Proof. Arrange 1, 2, . . . , n randomly around a circle. For each k-element subset of A of {1, 2, . . . , n},
we say that A is contiguous if all the elements of A lie in a contiguous block on the circle. The
probability that A forms a contiguous set on the circle is exactly
( n , since there are n possible
n
k)
positions for a continuous block of length k on the circle, and for each such choice, the probability
1
that A falls exactly into those position is . It follows that the expected number of continguous
( n k)
sets in F is exactly n|F|
( n k) .
□

4 YUFEI ZHAO
Since F is intersecting, there are at most k continguous sets in F. Indeed, suppose that A ∈ F
is contiguous. Then there are 2(k − 1) other contingous sets (not necessarily in F) that intersect A,
but they can be paired off into disjoint pairs. Since F is intersecting, it follows that it contains at
most k continguous sets. Combining with result from the previous paragraph, we see that n|F|
( n k) ≤ k,
and hence |F| ≤ n( k n
) (
k = n−1
k−1)
. □