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

Abstract

A graph is H-free if it has no induced subgraph isomorphic to H, and |G| denotes the number of vertices of G. A conjecture of Conlon, Sudakov and the second author asserts that: - For every graph H, there exists ∈ > 0 such that in every H-free graph G with |G| > 1 there are two disjoint sets of vertices, of sizes at least ∈|G|∈ and ∈|G|, complete or anticomplete to each other. This is equivalent to: - The "sparse linear conjecture": For every graph H, there exists ∈ > 0 such that in every H-free graph G with |G| > 1, either some vertex has degree at least ∈|G|, or there are two disjoint sets of vertices, of sizes at least ∈|G|∈ and ∈|G|, 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 that 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. (It remains open when H is the triangle K3.) There is also a stronger theorem: - For every almost-bipartite graph H, there exist ∈; t > 0 such that for every graph G with |G| > 1 and maximum degree less than ∈|G|, and for every c with 0 < c ≤ 1, either G contains ∈ct|G||H| induced copies of H, or there are two disjoint sets A,B V (G) with |A| ≥ ∈ct|G| and |B| ≥ ∈|G| and with at most c|A| . |B| edges between them. We also prove some variations on the sparse linear conjecture, such as: - For every graph H, there exists ∈ > 0 such that in every H-free graph G with |G| > 1 vertices, either some vertex has degree at least ∈|G|, or there are two disjoint sets A, B of vertices with |A| . |B| ≥ ∈|G|1+∈, anticomplete to each other.