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arXiv 1204.6595math.CO
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超图容器

Hypergraph containers

David Saxton, Andrew Thomason

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摘要：

我们发展了超图中独立集的包容性概念。对于每一个$r$ -均匀超图$G$，我们发现一个相对较小的收集$C$的顶点子集，使每一个独立的一组$G$包含在一个成员的$C$，没有成员的$C$是大的收集，这是在各个方面最佳的，揭示了一个底层结构的独立集。容器为许多与独立性有关的组合问题（通常是隐含的）提供了一种简单而统一的方法。关于着色，可以得出平均度为$d$的简单$r$ -一致超图的列表色数至少为$(1/(r-1)^2 + o(1)) \log_r d$ .对于$r = 2$，由于Alon，这改善了边界并且是紧的。对于$r \ge 3$，以前的边界是弱的，但目前的不等式是接近最优的。在极值图论的上下文中，由此得出，对于每个$\ell$ -阶为$k$的均匀超图$H$，存在$\ell$ -阶为$n$的均匀超图的集合$C$，每个都具有$H$的$o(n^k)$拷贝，使得每个无$H$的$\ell$-阶为$n$的均匀超图是$C$中超图的子图，和$\log |C| \le c n^{\ell-1/m(H)} \log n$，其中$m(H)$是标准参数（对于导出子图也有类似的声明）。这产生简单的证明，例如，一些$H$ -免费超图，并为稀疏定理的康隆-高尔斯和沙赫特。一个轻微的变体产生了KR猜想的计数版本。同样地，对于线性方程组，容器提供了例如无解集的数量的界限，以及稀疏随机子集中解的存在性。Balogh、Morris和Samotij独立地获得了相关结果。

英文摘要：

We develop a notion of containment for independent sets in hypergraphs. For every $r$-uniform hypergraph $G$, we find a relatively small collection $C$ of vertex subsets, such that every independent set of $G$ is contained within a member of $C$, and no member of $C$ is large; the collection, which is in various respects optimal, reveals an underlying structure to the independent sets. The containers offer a straightforward and unified approach to many combinatorial questions concerned (usually implicitly) with independence. With regard to colouring, it follows that simple $r$-uniform hypergraphs of average degree $d$ have list chromatic number at least $(1/(r-1)^2 + o(1)) \log_r d$. For $r = 2$ this improves a bound due to Alon and is tight. For $r \ge 3$, previous bounds were weak but the present inequality is close to optimal. In the context of extremal graph theory, it follows that, for each $\ell$-uniform hypergraph $H$ of order $k$, there is a collection $C$ of $\ell$-uniform hypergraphs of order $n$ each with $o(n^k)$ copies of $H$, such that every $H$-free $\ell$-uniform hypergraph of order $n$ is a subgraph of a hypergraph in $C$, and $\log |C| \le c n^{\ell-1/m(H)} \log n$ where $m(H)$ is a standard parameter (there is a similar statement for induced subgraphs). This yields simple proofs, for example, for the number of $H$-free hypergraphs, and for the sparsity theorems of Conlon-Gowers and Schacht. A slight variant yields a counting version of the KŁR conjecture. Likewise, for systems of linear equations the containers supply, for example, bounds on the number of solution-free sets, and the existence of solutions in sparse random subsets. Balogh, Morris and Samotij have independently obtained related results.

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