Ramsey-type problems on induced covers and induced partitions toward the Gyarfas-Sumner conjecture

Gyarfas and Sumner independently conjectured that for every tree T, there exists a function f(T) : n -> N such that every T-free graph G satisfies chi(G) <= f(T) (omega(G)), where chi(G) and omega(G) are the chromatic number and the clique number of G, respectively. This conjecture gives a solution of a Ramsey-type problem on the chromatic number. For a graph G, the induced SP-cover number inspc(G) (resp. the induced SP-partition number inspp(G)) of G is the minimum cardinality of a family P of induced subgraphs of G such that each element of P is a star or a path and boolean OR(P is an element of P) V (P) = V (G) (resp. (boolean OR) over dot(P is an element of P) V (P) = V (G)). 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 inspc and inspp, which are analogies of the Gyarfas-Sumner conjecture, and settle them. As a corollary of our results, we also settle other Ramsey-type problems for widely studied invariants.