纯对。VIII. 排除稀疏图
Pure pairs. VIII. Excluding a sparse graph
AI总结：
本文研究排除给定图及其补图作为诱导子图的图中纯对的最小尺寸，通过引入拥塞度概念，给出了大小为 n^{1-c} 的纯对存在与否的阈值条件。
摘要：
图 $G$ 中大小为 $t$ 的纯对是指两个互不相交的顶点集 $A,B$，每个集合包含 $t$ 个顶点，且 $A$ 与 $B$ 之间要么完全相邻，要么完全不相邻。已知对于每个森林 $H$，任意在 $n\ge2$ 个顶点上且不包含 $H$ 或其补图作为诱导子图的图，都具有大小为 $Ω(n)$ 的纯对；此外，这一性质仅在 $H$ 或其补图为森林时成立。本文研究大小为 $n^{1-c}$ 的纯对，其中 $0<c<1$。设 $H$ 为一个图：是否任意在 $n\ge2$ 个顶点上且不包含 $H$ 或其补图作为诱导子图的图，都具有纯对 $A,B$，满足 $|A|,|B|\ge Ω(|G|^{1-c})$？答案与 $H$ 的拥塞度有关，拥塞度定义为对 $H$ 的所有含边的子图 $J$ 取 $1-(|J|-1)/|E(J)|$ 的最大值。（拥塞度非负，且恰好当 $H$ 为森林时等于零。）设 $d$ 为 $H$ 与 $\overline{H}$ 的拥塞度中较小的一个。我们证明上述问题的答案在 $d\le c/(9+15c)$ 时为“是”，在 $d>c$ 时为“否”。
英文摘要：
A pure pair of size $t$ in a graph $G$ is a pair $A,B$ of disjoint sets of $t$ vertices such that $A$ is either complete or anticomplete to $B$. It is known that, for every forest $H$, every graph on $n\ge2$ vertices that does not contain $H$ or its complement as an induced subgraph has a pure pair of size $Ω(n)$; furthermore, this only holds when $H$ or its complement is a forest. In this paper, we look at pure pairs of size $n^{1-c}$, where $0<c<1$. Let $H$ be a graph: does every graph on $n\ge2$ vertices that does not contain $H$ or its complement as an induced subgraph have a pure pair $A,B$ with $|A|,|B|\ge Ω(|G|^{1-c})$,? The answer is related to the congestion of $H$, the maximum of $1-(|J|-1)/|E(J)|$ over all subgraphs $J$ of $H$ with an edge. (Congestion is nonnegative, and equals zero exactly when $H$ is a forest.) Let $d$ be the smaller of the congestions of $H$ and $\overline{H}$. We show that the answer to the question above is "yes" if $d\le c/(9+15c)$, and "no" if $d>c$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
