Sparse graphs with no polynomialsized anticomplete pairs
Abstract
A graph is "$H$free" if it has no induced subgraph isomorphic to $H$. A conjecture of Conlon, Fox and Sudakov states that for every graph $H$, there exists $s>0$ such that in every $H$free graph with $n>1$ vertices, either some vertex has degree at least $sn$, or there are two disjoint sets of vertices, of sizes at least $sn^s$ and $sn$, anticomplete to each other. We prove this holds for a large class of graphs $H$, and we prove that something like it holds for all graphs $H$. Say $H$ is "almostbipartite" if $H$ is trianglefree and $V(H)$ can be partitioned into a stable set and a set inducing a graph of maximum degree at most one. We prove that the conjecture above holds for when $H$ is almostbipartite. We also prove a stronger version where instead of excluding $H$ we restrict the number of copies of $H$. We prove some variations on the conjecture, such as: for every graph $H$, there exists $s >0$ such that in every $H$free graph with $n>1$ vertices, either some vertex has degree at least $sn$, or there are two disjoint sets $A, B$ of vertices with $AB > s n^{1 + s}$, anticomplete to each other.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1810.00058
 Bibcode:
 2018arXiv181000058C
 Keywords:

 Mathematics  Combinatorics