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Gy\'{a}rf\'{a}s and Sumner independently conjectured that for every tree $T$,there exists a function $f_{T}:\mathbb{N}\rightarrow \mathbb{N}$ such thatevery $T$-free graph $G$ satisfies $\chi (G)\leq f_{T}(\omega (G))$, where$\chi (G)$ and $\omega (G)$ are the {\it chromatic number} and the {\it cliquenumber} of $G$, respectively. This conjecture gives a solution of a Ramsey-typeproblem on the chromatic number.  For a graph $G$, the {\it induced SP-cover number ${\rm inspc}(G)$} (resp.the {\it induced SP-partition number ${\rm inspp}(G)$}) of $G$ is the minimumcardinality of a family $\mathcal{P}$ of induced subgraphs of $G$ such thateach element of $\mathcal{P}$ is a star or a path and $\bigcup _{P\in\mathcal{P}}V(P)=V(G)$ (resp. $\dot\bigcup _{P\in \mathcal{P}}V(P)=V(G)$). Suchtwo invariants are directly related concepts to the chromatic number. From theviewpoint of this fact, we focus on Ramsey-type problems for two invariants${\rm inspc}$ and ${\rm inspp}$, which are analogies of theGy\'{a}rf\'{a}s-Sumner conjecture, and settle them. As a corollary of ourresults, we also settle other Ramsey-type problems for widely studiedinvariants.

Ramsey-type problems on induced covers and induced partitions toward the Gy\'{a}rf\'{a}s-Sumner conjecture