arXiv 0901.20612009-01math.CO

染色H-自由超图

Coloring H-free Hypergraphs

Tom Bohman, Alan Frieze, Dhruv Mubayi

摘要：

修正$r \ge 2$和一组$r$ -均匀超图$\cH$。一个色数大于$k$的无$\cH$的$r$ -一致超图的最小边数是多少.我们调查这个问题的各种$\cH$。我们的结果包括：一个$(r,l)$ -系统是一个$r$ -一致超图，它的每两条边共享最多的$l$顶点。对于足够大的$k$，色数大于$k$的$(r,l)$ -系统中的最小边数至多为$c(k^{r-1}\log k)^{l/(l-1)}$，其中$$c<...$$这改进了以前的Kostochka-Mubayi-Rödl-Tetali \cite{KMRT}的最佳界。除了如\cite{KMRT}中所示的常数$c$之外，上限是尖锐的。具有独立邻域且色数大于$k$的$r$ -一致超图的最小边数是$\tilde k^{r+1/(r-1)}$阶为$k \to \infty$ .这推广（除了对数因子）的结果Gimbel和Gillassen \cite{GT}的三角形自由图。设$T$是$t$边的$r$ -一致超树.则每个$T$ -free $r$ -均匀超图的色数至多为$p(t)$，其中$p(t)$是$t$中的多项式.这推广了一个著名的事实，即每个$T$ -free图的色数至多为$t$ .还提出了几个开放的问题和建议。

英文摘要：

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-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.

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摘要：

英文摘要：

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