Coloring H-free Hypergraphs

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dc.creatorBohman, Tom
dc.creatorFrieze, Alan
dc.creatorMubayi, Dhruv
dc.date2009-01-14
dc.date2009-02-17
dc.date.accessioned2026-07-07T12:42:00Z
dc.date.available2026-07-07T12:42:00Z
dc.descriptionFix $r \ge 2$ 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-Rödl-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.
dc.identifierhttps://arxiv.org/abs/0901.2061
dc.identifierhttp://arxiv.org/abs/0901.2061
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/219935
dc.subjectCombinatorics
dc.subject05C65
dc.titleColoring H-free Hypergraphs
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