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 ε > 0 such that in every H-free graph G with |G| there are two disjoint sets of vertices, of sizes at least ε|G|ε and ε|G|, complete or anticomplete to each other. This is equivalent to: • The “sparse linear conjecture”: For every graph H, there exists e > 0 such that in every H-free graph G with |G| > 1, either some vertex has degree at least ε|G|, or there are two disjoint sets of vertices, of sizes at least ε|G|ε and ε|G|, 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 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 almostbipartite 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 ε, t > 0 such that for every graph G with |G| > 1 and maximum degree less than ε|G|, and for every c with 0 < c = 1, either G contains εct |G||H| induced copies of H, or there are two disjoint sets A, B ⊆ V(G) with |A| = ect|G| and |B| = e|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 ε > 0 such that in every H-free graph G with |G| > 1 vertices, either some vertex has degree at least ε|G|, or there are two disjoint sets A, B of vertices with |A|·|B| = ε|G|1+ε, anticomplete to each other.
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- Erdős-Hajnal conjecture
- induced subgraph