Combinatorics (Fall 2010)/Finite set systems

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Systems of Distinct Representatives (SDR)

Hall's marriage 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].

Doubly stochastic matrices

Theorem (Birkhoff 1949; von Neumann 1953)
Every doubly stochastic matrix is a convex combination of permutation matrices.

Cuckoo hashing

Min-max theorems

  • König-Egerváry theorem
  • Menger's theorem
  • Dilworth's theorem
Theorem (König 1931; Egerváry 1931)
In any bipartite graph, the size of a maximum matching equals the size of a minimum vertex cover.
Theorem (Menger 1927)
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 a[math]\displaystyle{ s }[/math]-[math]\displaystyle{ t }[/math] separating set.
Theorem (Dilworth 1950)
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.

Chains and antichains

Symmetric chains

Sperner's theorem

Hypergraph coloring

Theorem (Erdős 1963)
Let [math]\displaystyle{ \mathcal{F} }[/math] be a [math]\displaystyle{ k }[/math]-uniform. If [math]\displaystyle{ |\mathcal{F}|\lt 2^{k-1} }[/math] then [math]\displaystyle{ \mathcal{F} }[/math] is 2-colorable.

Lovász local lemma

Colorings

Theorem (Erdős-Lovász 1975)
Let [math]\displaystyle{ \mathcal{F} }[/math] be a [math]\displaystyle{ k }[/math]-uniform. If every member of [math]\displaystyle{ \mathcal{F} }[/math] intersects at most [math]\displaystyle{ 2^{k-3} }[/math] other members, then [math]\displaystyle{ \mathcal{F} }[/math] is 2-colorable.