Sparse graphs with no polynomial-sized anticomplete pairs

A graph is H-free if it has no induced subgraph isomorphic to H. An old conjecture of Conlon, Sudakov and the second author asserts that: • For every graph H, there exists ε > 0 such that in every H-free graph with n > 1 vertices there are two disjoint sets of vertices, of sizes at least εn and εn, complete or anticomplete to each other. This is equivalent to: • The ‘sparse linear conjecture’: For every graph H, there exists ε > 0 such that in every H-free graph with n > 1 vertices, either some vertex has degree at least εn, or there are two disjoint sets of vertices, of sizes at least εn and εn, anticomplete to each other. We prove a number of partial results towards the sparse linear conjecture. In particular, we prove it holds for a large class of graphs H, and we prove something like it holds for all graphs H. More exactly, say H is ‘almost-bipartite’ if H is triangle-free and V (H) 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 H. ∗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 Research Fellowship ‡Supported by ONR grant N00014-14-1-0084 and NSF grant DMS-1265563.