Pure pairs. III. Sparse graphs with no polynomial‐sized anticomplete pairs

Abstract

AbstractA graph is ‐free if it has no induced subgraph isomorphic to , and |G| denotes the number of vertices of . A conjecture of Conlon, Sudakov and the second author asserts that: For every graph , there exists such that in every ‐free graph with |G| there are two disjoint sets of vertices, of sizes at least and , complete or anticomplete to each other. This is equivalent to: The “sparse linear conjecture”: For every graph , there exists such that in every ‐free graph with , either some vertex has degree at least , or there are two disjoint sets of vertices, of sizes at least and , anticomplete to each other. We prove a number of partial results toward the sparse linear conjecture. In particular, we prove it holds for a large class of graphs , and we prove that something like it holds for all graphs . More exactly, say is “almost‐bipartite” if is triangle‐free and can be partitioned into a stable set and a set inducing a graph of maximum degree at most one. (This includes all graphs that arise from another graph by subdividing every edge at least once.) Our main result is: The sparse linear conjecture holds for all almost‐bipartite graphs . (It remains open when is the triangle .) There is also a stronger theorem: For every almost‐bipartite graph , there exist such that for every graph with and maximum degree less than , and for every with , either contains induced copies of , or there are two disjoint sets with and , and with at most edges between them. We also prove some variations on the sparse linear conjecture, such as: For every graph , there exists such that in every ‐free graph with vertices, either some vertex has degree at least , or there are two disjoint sets of vertices with , anticomplete to each other.