Excluding a Substar and an Antisubstar

Abstract

Ramsey's theorem says that for every clique $H_1$ and for every graph $H_2$ with no edges, all graphs containing neither of $H_1,H_2$ as induced subgraphs have bounded order. What if, instead, we exclude a graph $H_1$ with a vertex whose deletion gives a clique, and the complement $H_2$ of another such graph? This no longer implies bounded order, but it implies tightly restricted structure that we describe. There are also several related subproblems (what if we exclude a star and the complement of a star? what if we exclude a star and a clique? and so on) and we answer a selection of these.