PROBABILISTIC METHODS IN COMBINATORICS 1. Introduction ...

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) .
□