We prove for every graph H there exists ɛ > 0 such that, for every graph G with |G|≥2, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least ɛ|G| neighbours, or there are two disjoint sets A, B ⊆ V(G) with |A|,|B|≥ɛ|G| such that no edge joins A and B. It follows that for every graph H, there exists c>0 such that for every graph G, if no induced subgraph of G or its complement is a subdivision of H, then G has a clique or stable set of cardinality at least |G|^c. This is related to the Erdős-Hajnal conjecture.

中文翻译：

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纯对。二、排除图的所有细分

我们证明对于每个图H 都存在 ɛ > 0 使得对于每个图G具有 | G |≥2，如果G 的诱导子图没有是H的细分，则G 的某个顶点至少有 ɛ| G | 邻居，或者有两个不相交的集合A , B ⊆ V ( G ) 与 | 一个|,| B |≥ɛ| G | 使得没有边连接A和B。因此对于每个图H，都存在c >0 使得对于每个图ģ，如果没有导出子ģ或其互补是细分ħ，然后ģ具有集团或稳定组基数的至少| G | ^丙。这与 Erdős-Hajnal 猜想有关。