825 research outputs found

    Ramsey-nice families of graphs

    Get PDF
    For a finite family F\mathcal{F} of fixed graphs let Rk(F)R_k(\mathcal{F}) be the smallest integer nn for which every kk-coloring of the edges of the complete graph KnK_n yields a monochromatic copy of some FFF\in\mathcal{F}. We say that F\mathcal{F} is kk-nice if for every graph GG with χ(G)=Rk(F)\chi(G)=R_k(\mathcal{F}) and for every kk-coloring of E(G)E(G) there exists a monochromatic copy of some FFF\in\mathcal{F}. It is easy to see that if F\mathcal{F} contains no forest, then it is not kk-nice for any kk. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs F\mathcal{F} that contains at least one forest, and for all kk0(F)k\geq k_0(\mathcal{F}) (or at least for infinitely many values of kk), F\mathcal{F} is kk-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 F\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.Comment: 20 pages, 2 figure

    The history of degenerate (bipartite) extremal graph problems

    Full text link
    This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

    Entropy inequalities for factors of iid

    Get PDF
    This paper is concerned with certain invariant random processes (called factors of IID) on infinite trees. Given such a process, one can assign entropies to different finite subgraphs of the tree. There are linear inequalities between these entropies that hold for any factor of IID process (e.g. "edge versus vertex" or "star versus edge"). These inequalities turned out to be very useful: they have several applications already, the most recent one is the Backhausz-Szegedy result on the eigenvectors of random regular graphs. We present new entropy inequalities in this paper. In fact, our approach provides a general "recipe" for how to find and prove such inequalities. Our key tool is a generalization of the edge-vertex inequality for a broader class of factor processes with fewer symmetries
    corecore