Ramsey-nice families of graphs

Abstract

For a finite family $\cF$ of fixed graphs let $R_k(\cF)$ 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 $F\in\cF$. We say that $\cF$ is \emph{$k$-nice} if for every graph $G$ with $\chi(G)=R_k(\cF)$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\cF$.

It is easy to see that if $\cF$ 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 $\cF$ that contains at least one forest, and for all $k\geq k_0(\cF)$ (or at least for infinitely many values of $k$), $\cF$ 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 $\cF$ 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.