The Erdos-Hajnal Conjecture asserts that for every graph H there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of cardinality at least |G|^c. In this paper, we prove a conjecture of Liebenau, Pilipczuk, and the last two authors, that for every forest H there exists c > 0, such that every graph G contains either an induced copy of H, or a vertex of degree at least c|G|, or two disjoint sets of at least c|G| vertices with no edges between them. It follows that for every forest H there is c > 0 so that if G contains neither H nor its complement as an induced subgraph then there is a clique or stable set of cardinality at least |G|^c.

中文翻译：

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纯对。一、树和线性反完全对

Erdos-Hajnal 猜想断言，对于每个图 H，都有一个常数 c > 0，使得每个不包含 H 作为诱导子图的图 G 至少具有 |G|^c 的团或稳定基数集。在本文中，我们证明了 Liebenau、Pilipczuk 和最后两位作者的一个猜想，即对于每个森林 H 都存在 c > 0，这样每个图 G 要么包含 H 的诱导副本，要么至少包含一个度数的顶点c|G|，或至少 c|G| 的两个不相交的集合 顶点之间没有边。因此，对于每个森林 H，都有 c > 0，因此如果 G 既不包含 H 也不包含其补集作为诱导子图，那么至少有一个集团或稳定的基数集 |G|^c。