Coloring H-free Hypergraphs

TL;DR

This paper investigates the minimum edges needed in hypergraphs avoiding certain substructures while having high chromatic number, providing new bounds and generalizations for various hypergraph classes.

Contribution

It introduces improved bounds for the minimum edges in hypergraphs with forbidden configurations and high chromatic number, extending classical results to hypergraphs.

Findings

Bound for $(r,l)$-systems with high chromatic number improves previous results.

Minimum edges in hypergraphs with independent neighborhoods grow as ilde{k}^{r+1/(r-1)}.

Chromatic number of $T$-free hypergraphs is polynomial in the size of the hypertree.

Abstract

Fix $r \ge 2$ and a collection of $r$-uniform hypergraphs $\cH$. What is the minimum number of edges in an $\cH$-free $r$-uniform hypergraph with chromatic number greater than $k$. We investigate this question for various $\cH$. Our results include the following: An $(r,l)$-system is an $r$-uniform hypergraph with every two edges sharing at most $l$ vertices. For $k$ sufficiently large, the minimum number of edges in an $(r,l)$-system with chromatic number greater than $k$ is at most $c(k^{r-1}\log k)^{l/(l-1)}$, where $$c<...$$ This improves on the previous best bounds of Kostochka-Mubayi-R\"odl-Tetali \cite{KMRT}. The upper bound is sharp aside from the constant $c$ as shown in \cite{KMRT}. The minimum number of edges in an $r$-uniform hypergraph with independent neighborhoods and chromatic number greater than $k$ is of order $\tilde k^{r+1/(r-1)}$ as $k \to \infty$. This generalizes (aside from logarithmic factors) a result of Gimbel and Thomassen \cite{GT} for triangle-free graphs. Let $T$ be an $r$-uniform hypertree of $t$ edges. Then every $T$-free $r$-uniform hypergraph has chromatic number at most $p(t)$, where $p(t)$ is a polynomial in $t$. This generalizes the well known fact that every $T$-free graph has chromatic number at most $t$. Several open problems and conjectures are also posed.