Coloring hypergraphs with bounded cardinalities of edge intersections

TL;DR

This paper proves that certain hypergraphs with bounded edge intersections and degrees can be colored with a limited number of colors, extending understanding of hypergraph colorability under intersection constraints.

Contribution

It establishes a new bound on the colorability of $b$-simple hypergraphs with bounded edge degrees, generalizing previous results in hypergraph coloring theory.

Findings

Hypergraphs with bounded edge degrees are $r$-colorable under specified conditions.

The result applies to $n$-uniform $b$-simple hypergraphs for sufficiently large $n$.

Provides applications of the main coloring theorem.

Abstract

The paper deals with an extremal problem concerning colorings of hypergraphs with bounded edge degrees. Consider the family of $b$-simple hypergraphs, in which any two edges do not share more than $b$ common vertices. We prove that for $n\geqslant n_0(b)$, any $n$-uniform $b$-simple hypergraph with the maximum edge degree at most $c\cdot nr^{n-b}$ is $r$-colorable, where $c>0$ is an absolute constant. We also establish some applications of the main result.