Combinatorics (Fall 2010)/Extremal set theory II

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.

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.