A “cap” in a graph G is an induced subgraph of G that consists of a cycle of length at least four, together with one further vertex that has exactly two neighbours in the cycle, adjacent to each other, and the “house” is the smallest, on five vertices. It is not known whether there exists $\epsilon >0$ such that every graph G containing no house has a clique or stable set of cardinality at least $|G{|}^{\epsilon }$; this is the smallest open case of the Erdős-Hajnal conjecture and has been the subject of much study.

We prove that there exists $\epsilon >0$ such that every graph G with no cap has a clique or stable set of cardinality at least $|G{|}^{\epsilon }$.

中文翻译：

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Erdős-Hajnal 用于无上限图

A“帽”中的曲线图G ^是一个导出子ģ与一个另外的顶点，其具有在周期正好两个邻居，彼此相邻的由长度的周期的至少四个，一起，和“房子”是最小的，在五个顶点上。不知道是否存在$\epsilon >0$使得每个不包含房子的图G至少有一个集团或稳定的基数集$|G{|}^{\epsilon }$; 这是 Erdős-Hajnal 猜想的最小开放案例，并且一直是许多研究的主题。

我们证明存在 $\epsilon >0$使得每个没有上限的图G至少有一个集团或稳定的基数集$|G{|}^{\epsilon }$.