Ramsey‐type problems on induced covers and induced partitions toward the Gyárfás–Sumner conjecture

Abstract

Gyárfás and Sumner independently conjectured that for every tree , there exists a function such that every ‐free graph satisfies , where and are the chromatic number and the clique number of , respectively. This conjecture gives a solution of a Ramsey‐type problem on the chromatic number. For a graph , the induced SP‐cover number (resp. the induced SP‐partition number ) of is the minimum cardinality of a family of induced subgraphs of such that each element of is a star or a path and (resp. ). Such two invariants are directly related concepts to the chromatic number. From the viewpoint of this fact, we focus on Ramsey‐type problems for two invariants and , which are analogies of the Gyárfás‐Sumner conjecture, and settle them. As a corollary of our results, we also settle other Ramsey‐type problems for widely studied invariants.