Coloring H-free Hypergraphs

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.