Combinatorics (Fall 2010)/Finite set systems
Systems of Distinct Representatives (SDR)
Hall's marriage theorem
Hall's Theorem (SDR) - The sets [math]\displaystyle{ S_1,S_2,\ldots,S_m }[/math] have a system of distinct representatives (SDR) if and only if
- [math]\displaystyle{ \left|\bigcup_{i\in I}S_i\right|\ge |I| }[/math] for all [math]\displaystyle{ I\subseteq\{1,2,\ldots,m\} }[/math].
- The sets [math]\displaystyle{ S_1,S_2,\ldots,S_m }[/math] have a system of distinct representatives (SDR) if and only if
Hall's Theorem (matching in bipartite graph) - A bipartite graph [math]\displaystyle{ G(U,V,E) }[/math] has a matching of [math]\displaystyle{ U }[/math] if and only if
- [math]\displaystyle{ \left|N(S)\right|\ge |S| }[/math] for all [math]\displaystyle{ S\subseteq U }[/math].
- A bipartite graph [math]\displaystyle{ G(U,V,E) }[/math] has a matching of [math]\displaystyle{ U }[/math] if and only if
Doubly stochastic matrices
Theorem (Birkhoff 1949; von Neumann 1953) - Every doubly stochastic matrix is a convex combination of permutation matrices.
Min-max theorems
- König-Egerváry theorem (König 1931; Egerváry 1931): in a bipartite graph, the maximum number of edges in a matching equals the minimum number of vertices in a vertex cover.
- Menger's theorem (Menger 1927): the minimum number of vertices separating two given vertices in a graph equals the maximum number of vertex-disjoint paths between the two vertices.
- Dilworth's theorem (Dilworth 1950): the minimum number of chains which cover a partially ordered set equals the maximum number of elements in an antichain.
König-Egerváry Theorem (graph theory form) - In any bipartite graph, the size of a maximum matching equals the size of a minimum vertex cover.
König-Egerváry Theorem (matrix form) - Let [math]\displaystyle{ A }[/math] be an [math]\displaystyle{ m\times n }[/math] 0-1 matrix. The maximum number of independent 1's is equal to the minimum number of rows and columns required to cover all the 1's in [math]\displaystyle{ A }[/math].
Menger's Theorem - Let [math]\displaystyle{ G }[/math] be a graph and let [math]\displaystyle{ s }[/math] and [math]\displaystyle{ t }[/math] be two vertices of [math]\displaystyle{ G }[/math]. The maximum number of internally disjoint paths from [math]\displaystyle{ s }[/math] to [math]\displaystyle{ t }[/math] equals the minimum number of vertices in an [math]\displaystyle{ s }[/math]-[math]\displaystyle{ t }[/math] separating set.
Chains and antichains
Dilworth's theorem
Dilworth's Theorem - Suppose that the largest antichain in the poset [math]\displaystyle{ P }[/math] has size [math]\displaystyle{ r }[/math]. Then [math]\displaystyle{ P }[/math] can be partitioned into [math]\displaystyle{ r }[/math] chains.
Application: Erdős-Szekeres Theorem
Erdős-Szekeres Theorem - A sequence of more than [math]\displaystyle{ mn }[/math] different real numbers must contain either an increasing subsequence of length [math]\displaystyle{ m+1 }[/math], or a decreasing subsequence of length [math]\displaystyle{ n+1 }[/math].
Proof by Dilworth's theorem (Original proof of Erdős-Szekeres)
- [math]\displaystyle{ \square }[/math]
Application: Hall's Theorem
Hall's Theorem - The sets [math]\displaystyle{ S_1,S_2,\ldots,S_m }[/math] have a system of distinct representatives (SDR) if and only if
- [math]\displaystyle{ \left|\bigcup_{i\in I}S_i\right|\ge |I| }[/math] for all [math]\displaystyle{ I\subseteq\{1,2,\ldots,m\} }[/math].
- The sets [math]\displaystyle{ S_1,S_2,\ldots,S_m }[/math] have a system of distinct representatives (SDR) if and only if
Proof by Dilworth's theorem - [math]\displaystyle{ \square }[/math]
Symmetric chain decomposition
Theorem (de Bruijn et al 1952) - [math]\displaystyle{ 2^S }[/math] with [math]\displaystyle{ |S|=n }[/math] can be partitioned into at most [math]\displaystyle{ {n\choose \lfloor n/2\rfloor} }[/math] mutually disjoint symmetric chains.
Application: the memory allocation problem
Sperner system
Theorem (Sperner 1928) - Let [math]\displaystyle{ |S|=n }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq 2^S }[/math] be an antichain. Then
- [math]\displaystyle{ |\mathcal{F}|\le{n\choose \lfloor n/2\rfloor} }[/math].
- Let [math]\displaystyle{ |S|=n }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq 2^S }[/math] be an antichain. Then
First proof (symmetric chain decomposition)
Proof of Sperner's theorem - [math]\displaystyle{ \square }[/math]
Second proof (shadowing)
Definition - Let [math]\displaystyle{ |S|=n\, }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq {S\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\{A\in {S\choose k+1}\,\,\bigg|\,\, \exists B\in\mathcal{F}\mbox{ such that } B\subset A\right\} }[/math].
- Thus the shade [math]\displaystyle{ \nabla\mathcal{F} }[/math] of [math]\displaystyle{ \mathcal{F} }[/math] consists of all subsets of [math]\displaystyle{ S }[/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\{A\in {S\choose k+1}\,\,\bigg|\,\, \exists B\in\mathcal{F}\mbox{ such that } B\subset A\right\} }[/math].
- Thus the shadow [math]\displaystyle{ \Delta\mathcal{F} }[/math] of [math]\displaystyle{ \mathcal{F} }[/math] consists of all subsets of [math]\displaystyle{ S }[/math] which can be obtained by removing an element from a set in [math]\displaystyle{ \mathcal{F} }[/math].
Lemma (Sperner) - Let [math]\displaystyle{ |S|=n\, }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq {S\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]
- Let [math]\displaystyle{ |S|=n\, }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq {S\choose k} }[/math]. Then
Proof of Sperner's theorem (original proof of Sperner)
- [math]\displaystyle{ \square }[/math]
Third 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{ |S|=n }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq 2^S }[/math] be an antichain. For [math]\displaystyle{ k=0,1,\ldots,n }[/math], let [math]\displaystyle{ f_k=|\{A\in\mathcal{F}\mid |A|=k\}| }[/math]. Then
- [math]\displaystyle{ \sum_{A\in\mathcal{F}}\frac{1}{{n\choose |A|}}=\sum_{k=0}^n\frac{f_k}{{n\choose k}}\le 1 }[/math].
- Let [math]\displaystyle{ |S|=n }[/math] and [math]\displaystyle{ \mathcal{F}\subseteq 2^S }[/math] be an antichain. For [math]\displaystyle{ k=0,1,\ldots,n }[/math], let [math]\displaystyle{ f_k=|\{A\in\mathcal{F}\mid |A|=k\}| }[/math]. Then
Another proof (the probabilistic method) - [math]\displaystyle{ \square }[/math]