Extremal results on hypergraphs arising from geometric objects

Abstract

A graph class $\mathcal{G}$ has the strong Erd\H{o}s-Hajnal(EH) property if there is a constant $c=c(\mathcal{G}) > 0$ such that for every member $G$ of $\mathcal{G}$, either $G$ or its complement has $K_{m, m}$ as a subgraph where $m \geq \lfloor c |V(G)| \rfloor$. We prove that the class of chordal graphs satisfy the strong Erd\H{o}s-Hajnal property with constant $c = \frac{2}{9}$. A strengthening of the strong EH property which we call the colorful strong EH property was discussed in geometric settings by Alon et al. and by Fox et al. Inspired by their results, we disprove the colorful strong EH property of chordal graphs and intersection graphs of convex sets on the plane. Erd\H{o}s-Hajnal type properties for other related graph classes are also discussed. In the next topic, we consider the transversal number of facet hypergraphs of a certain family of polytopes. It is conjectured that every facet hypergraph of a convex polytope can be pierced by at most half of its vertices. We provide a new lower bound for the transversal number of stacked 2-spheres which are exactly facet hypergraphs of stacked 3-polytopes. Finally, we show the existence of rainbow matchings in bipartite graphs under a cooperative condition on color classes. Our proof is purely combinatorial. Holmsen and Lee obtained the same result using topological properties of certain graph complexes which is also enjoyed by the nerve of convex sets. Using those properties, we draw relations between our result and cooperative theorems in discrete geometry.