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].

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].

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

Symmetric chains

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.

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].