Coloring hypergraphs that are the union of nearly disjoint cliques

TL;DR

This paper investigates the chromatic number of hypergraphs formed by nearly disjoint cliques, introducing new constructions with polynomially many cliques and analyzing their coloring properties, including bounds in geometric configurations.

Contribution

It presents novel polynomial-sized near design constructions for hypergraphs with large chromatic number and analyzes their properties using probabilistic methods.

Findings

Constructed hypergraphs with polynomially many cliques and high chromatic number.

Provided lower bounds on chromatic number using a random greedy process.

Derived bounds on the maximum number of caps in finite projective/affine planes.

Abstract

We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are known to exist only when the number of cliques is exponential in the clique size. We construct near designs where the number of cliques is polynomial in the clique size, and show that they have large chromatic number. The case when the cliques have pairwise intersections of size at most one seems particularly challenging. Here we give lower bounds by analyzing a random greedy hypergraph process. We also consider the related question of determining the maximum number of caps in a finite projective/affine plane and obtain nontrivial upper and lower bounds.