Combinatorics (Fall 2010)/Extremal set theory

Sperner system

Theorem (Sperner 1928)

Let [math]\displaystyle{ |X|=n }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq 2^X }[/math] be an antichain. Then [math]\displaystyle{ |\mathcal{F}|\le{n\choose \lfloor n/2\rfloor} }[/math].

First proof (shadowing)

Definition

Let [math]\displaystyle{ |X|=n\, }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq {X\choose k} }[/math], [math]\displaystyle{ k\lt n\, }[/math]. The shade of [math]\displaystyle{ \mathcal{F} }[/math] is defined to be [math]\displaystyle{ \nabla\mathcal{F}=\left\{T\in {X\choose k+1}\,\,\bigg|\,\, \exists S\in\mathcal{F}\mbox{ such that } S\subset T\right\} }[/math]. Thus the shade [math]\displaystyle{ \nabla\mathcal{F} }[/math] of [math]\displaystyle{ \mathcal{F} }[/math] consists of all subsets of [math]\displaystyle{ X }[/math] which can be obtained by adding an element to a set in [math]\displaystyle{ \mathcal{F} }[/math]. Similarly, the shadow of [math]\displaystyle{ \mathcal{F} }[/math] is defined to be [math]\displaystyle{ \Delta\mathcal{F}=\left\{T\in {X\choose k-1}\,\,\bigg|\,\, \exists S\in\mathcal{F}\mbox{ such that } T\subset S\right\} }[/math]. Thus the shadow [math]\displaystyle{ \Delta\mathcal{F} }[/math] of [math]\displaystyle{ \mathcal{F} }[/math] consists of all subsets of [math]\displaystyle{ X }[/math] which can be obtained by removing an element from a set in [math]\displaystyle{ \mathcal{F} }[/math].

Lemma (Sperner)

Let [math]\displaystyle{ |X|=n\, }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq {X\choose k} }[/math]. Then [math]\displaystyle{ \begin{align} &|\nabla\mathcal{F}|\ge\frac{n-k}{k+1}|\mathcal{F}| &\text{ if } k\lt n\\ &|\Delta\mathcal{F}|\ge\frac{k}{n-k+1}|\mathcal{F}| &\text{ if } k\gt 0. \end{align} }[/math]

Proof of Sperner's theorem (original proof of Sperner) [math]\displaystyle{ \square }[/math]

Second proof (double counting)

Proof of Sperner's theorem (Lubell 1966) [math]\displaystyle{ \square }[/math]

The LYM inequality

Theorem (Lubell, Yamamoto 1954; Meschalkin 1963)

Let [math]\displaystyle{ |X|=n }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq 2^X }[/math] be an antichain. For [math]\displaystyle{ k=0,1,\ldots,n }[/math], let [math]\displaystyle{ f_k=|\{S\in\mathcal{F}\mid |S|=k\}| }[/math]. Then [math]\displaystyle{ \sum_{S\in\mathcal{F}}\frac{1}{{n\choose |S|}}=\sum_{k=0}^n\frac{f_k}{{n\choose k}}\le 1 }[/math].

Third proof (the probabilistic method) (Due to Alon.) [math]\displaystyle{ \square }[/math]

Proposition

[math]\displaystyle{ \sum_{S\in\mathcal{F}}\frac{1}{{n\choose |S|}}\le 1 }[/math] implies that [math]\displaystyle{ |\mathcal{F}|\le{n\choose \lfloor n/2\rfloor} }[/math].

Intersecting Families

Sunflowers

Sunflower Lemma

Let [math]\displaystyle{ \mathcal{F}\subseteq {X\choose k} }[/math]. If [math]\displaystyle{ |\mathcal{F}|\gt k!(r-1)^k }[/math], then [math]\displaystyle{ \mathcal{F} }[/math] contains a sunflower of size [math]\displaystyle{ r }[/math].

Erdős–Ko–Rado theorem

Erdős–Ko–Rado theorem

Let [math]\displaystyle{ \mathcal{F}\subseteq {X\choose k} }[/math]. If for any [math]\displaystyle{ S,T\in\mathcal{F} }[/math], [math]\displaystyle{ S\cap T\neq\emptyset }[/math], then [math]\displaystyle{ |\mathcal{F}|\le{n-1\choose k-1} }[/math].

Kruskal–Katona theorem

Theorem (Kruskal 1963, Katona 1966)

Let [math]\displaystyle{ \mathcal{F}\subseteq {X\choose k} }[/math] with [math]\displaystyle{ |\mathcal{F}|=m }[/math], and suppose that [math]\displaystyle{ m={n_k\choose k}+{n_{k-1}\choose k-1}+\cdots+{n_t\choose t} }[/math] where [math]\displaystyle{ a_k\gt a_{k-1}\gt \cdots\gt a_t\ge t\ge 1 }[/math]. Then [math]\displaystyle{ |\Delta\mathcal{F}|\ge {n_k\choose k-1}+{n_{k-1}\choose k-2}+\cdots+{n_t\choose t-1} }[/math].