Ramsey-nice families of graphs

Abstract

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.