Ramsey-nice families of graphs
Ron Aharoni, Noga Alon, Michal Amir, Penny Haxell, Dan Hefetz, Zilin, Jiang, Gal Kronenberg, and Alon Naor

TL;DR
This paper investigates conditions under which families of graphs guarantee monochromatic subgraphs in edge colorings of complete graphs, proposing a conjecture and providing partial results supporting it.
Contribution
It introduces the concept of $k$-nice families of graphs, conjectures a characterization involving forests, and proves several cases supporting this conjecture.
Findings
Confirmed $k$-niceness for families with two connected graphs of 3 edges.
Established $k$-niceness for families containing a forest with at most 2 edges.
Disproved a related conjecture on matchings in 3-uniform hypergraphs.
Abstract
For a finite family of fixed graphs let be the smallest integer for which every -coloring of the edges of the complete graph yields a monochromatic copy of some . We say that is -nice if for every graph with and for every -coloring of there exists a monochromatic copy of some . It is easy to see that if contains no forest, then it is not -nice for any . It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs that contains at least one forest, and for all (or at least for infinitely many values of ), is -nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the…
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