A triangle in a hypergraph $\mathcal{H}$ is a set of three distinct edges $e, f, g\in\mathcal{H}$ and three distinct vertices $u, v, w\in V(\mathcal{H})$ such that $\{u, v\}\subseteq e$, $\{v, w\}\subseteq f$, $\{w, u\}\subseteq g$ and $\{u, v, w\}\cap e\cap f\cap g=\emptyset$. Johansson proved in 1996 that $\chi(G)=\mathcal{O}(\Delta/\log\Delta)$ for any triangle-free graph $G$ with maximum degree $\Delta$. Cooper and Mubayi later generalized the Johansson's theorem to all rank $3$ hypergraphs. In this paper we provide a common generalization of both these results for all hypergraphs, showing that if $\mathcal{H}$ is a rank $k$, triangle-free hypergraph, then the list chromatic number \[ \chi_{\ell}(\mathcal{H})\leq \mathcal{O}\left(\max_{2\leq \ell \leq k} \left\{\left( \frac{\Delta_{\ell}}{\log \Delta_{\ell}} \right)^{\frac{1}{\ell-1}} \right\}\right), \] where $\Delta_{\ell}$ is the maximum $\ell$-degree of $\mathcal{H}$. The result is sharp apart from the constant. Moreover, our result implies, generalizes and improves several earlier results on the chromatic number and also independence number of hypergraphs, while its proof is based on a different approach than prior works in hypergraphs (and therefore provides alternative proofs to them). In particular, as an application, we establish a bound on chromatic number of sparse hypergraphs in which each vertex is contained in few triangles, and thus extend results of Alon, Krivelevich and Sudakov, and Cooper and Mubayi from hypergraphs of rank 2 and 3, respectively, to all hypergraphs.
翻译:高音中的三角形 $\ mathcal{H} 美元是三種不同的邊緣 $, f, g\ nin\ mathcal{H} 美元, 和三種不同的脊椎 $u, v\\ mathcal{H} 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 立方 立方 美元, 立方 立方 美元, 立方 立方 美元, 立方美元, 立方 立方美元, 立方美元, 库珀和穆拜依次, 将约翰森的方言通至所有级 3美元。 在本文中,我们对所有高音都提供了上述结果的通用概略化, 显示如果美元 和直立方美元, 立方言中, 立方言中以美元为美元, 立方美元, 立方言中以美元, 立方方方方言 立方言方言以xx=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx