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dc.contributor.authorAharoni, Ron
dc.contributor.authorAlon, Noga
dc.contributor.authorAmir, Michal
dc.contributor.authorHaxell, Penny
dc.contributor.authorHefetz, Dan
dc.contributor.authorJiang, Zilin
dc.contributor.authorKronenberg, Gal
dc.contributor.authorNaor, Alon 14:50:10 (GMT) 14:50:10 (GMT)
dc.description© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
dc.description.abstractFor 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.en
dc.description.sponsorshipResearch supported in part by a BSF grant, an ISF grant and a GIF grant.en
dc.rightsAttribution-NonCommercial-NoDerivatives 4.0 International*
dc.titleRamsey-nice families of graphsen
dcterms.bibliographicCitationAharoni, Ron, Noga Alon, Michal Amir, Penny Haxell, Dan Hefetz, Zilin Jiang, Gal Kronenberg, and Alon Naor. ‘Ramsey-Nice Families of Graphs’. European Journal of Combinatorics 72 (1 August 2018): 29–44.
uws.contributor.affiliation1Faculty of Mathematicsen
uws.contributor.affiliation2Combinatorics and Optimizationen

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