About dependence of the number of edges and vertices in hypergraph clique with chromatic number 3

TL;DR

This paper investigates the relationship between the number of edges and vertices in hypergraph cliques with chromatic number 3, improving existing bounds when the number of vertices is limited by a function of the uniformity.

Contribution

It provides a new upper bound on the number of edges in n-uniform hypergraphs with bounded vertices and chromatic number 3, extending previous results.

Findings

Improved upper bound on edges for hypergraphs with bounded vertices.

Extension of Erdős-Lovász bound for specific vertex constraints.

Refinement of bounds when vertices are limited by a function of n.

Abstract

In 1973 P. Erd\H{o}s and L. Lov\'asz noticed that any hypergraph whose edges are pairwise intersecting has chromatic number 2 or 3. In the first case, such hypergraph may have any number of edges. However, Erd\H{o}s and Lov\'asz proved that in the second case, the number of edges is bounded from above. For example, if a hypergraph is $ n $-uniform, has pairwise intersecting edges, and has chromatic number 3, then the number of its edges does not exceed $ n^n $. Recently D.D. Cherkashin improved this bound (see \cite{Ch}). In this paper, we further improve it in the case when the number of vertices of an $n$-uniform hypergraph is bounded from above by $ n^m $ with some $ m = m(n) $.