  For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ 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 $F\in\mathcal{F}$. We
say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with
$\chi(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a
monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if
$\mathcal{F}$ 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 $\mathcal{F}$ that contains at least one forest, and for all
$k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$),
$\mathcal{F}$ 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 $\mathcal{F}$ containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.
