论文标题
整个地方都有大集团和独立套装
Large cliques and independent sets all over the place
论文作者
论文摘要
我们研究了90年代初期Erdős和Hajnal提出的以下问题。在所有$ n $ vertex图中，$ g $什么是$ m $的最小可能值，其中任何$ m $ g $的$ m $顶点都包含一个集团和一组独立的尺寸$ \ log n $？我们构建示例表明，$ m $最多是$ 2^{2^{（\ log \ log \ log n）^{1/2+o（1）}}}} $，获得了大约$ \ sqrt {n} $的上限的双重级别级别改进。我们的（概率）结构引起了拉姆齐图的新示例，尽管没有很大的同质子集在任何小部分的顶点中都包含群集和大小$ \ log n $的独立集。在随机图中，这与真实相去甚远。我们的证明是基于服用词典产物和使用随机性之间的相互作用的。
We study the following question raised by Erdős and Hajnal in the early 90's. Over all $n$-vertex graphs $G$ what is the smallest possible value of $m$ for which any $m$ vertices of $G$ contain both a clique and an independent set of size $\log n$? We construct examples showing that $m$ is at most $2^{2^{(\log\log n)^{1/2+o(1)}}}$ obtaining a twofold sub-polynomial improvement over the upper bound of about $\sqrt{n}$ coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size $\log n$ in any small subset of vertices. This is very far from being true in random graphs. Our proofs are based on an interplay between taking lexicographic products and using randomness.
