Ramsey-nice families of graphs

For a finite family F of fixed graphs let R_k(F) be the smallest integer n for which every k-coloring of the edges of the complete graph K_n yields a monochromatic copy of some F∈F. We say that F is k-nice if for every graph G with χ(G)=R_k(F) and for every k-coloring of E(G) there exists a monochromatic copy of some F∈F. It is easy to see that if F contains no forest, then it is not k-nice for any k. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs F that contains at least one forest, and for all k≥k₀(F) (or at least for infinitely many values of k), F is k-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family F containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni et al. (2015) regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.