Pure pairs. I. Trees and linear anticomplete pairs

The final publication is available at Elsevier via https://doi.org/10.1016/j.aim.2020.107396. © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The Erdős-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 and Pilipczuk [11], that for every forest H there exists c > 0, such that every graph G with |G| > 1 contains either an induced copy of H, or a vertex of degree at least c|Gj|, or two disjoint sets of at least c|G| vertices with no edges between them. It follows that for every forest H there exists c > 0 such 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.Supported by NSF grant DMS-1550991. This material is based upon work supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office under grant number W911NF-16-1-0404. Supported by a Leverhulme Trust Research Fellowship. Supported by ONR grant N00014-14-1-0084, and NSF grant DMS-1265563, and AFOSR grant A9550-19-1-0187. This work was mostly performed while Spirkl was at Princeton University