352 research outputs found

    A Folkman Linear Family

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    For graphs FF and GG, let Fβ†’(G,G)F\to (G,G) signify that any red/blue edge coloring of FF contains a monochromatic GG. Define Folkman number f(G;p)f(G;p) to be the smallest order of a graph FF such that Fβ†’(G,G)F\to (G,G) and Ο‰(F)≀p\omega(F) \le p. It is shown that f(G;p)≀cnf(G;p)\le cn for graphs GG of order nn with Ξ”(G)≀Δ\Delta(G)\le \Delta, where Ξ”β‰₯3\Delta\ge 3, c=c(Ξ”)c=c(\Delta) and p=p(Ξ”)p=p(\Delta) are positive constants.Comment: 11 page

    Combinatorial theorems relative to a random set

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    We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

    Improved bounds on the multicolor Ramsey numbers of paths and even cycles

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    We study the multicolor Ramsey numbers for paths and even cycles, Rk(Pn)R_k(P_n) and Rk(Cn)R_k(C_n), which are the smallest integers NN such that every coloring of the complete graph KNK_N has a monochromatic copy of PnP_n or CnC_n respectively. For a long time, Rk(Pn)R_k(P_n) has only been known to lie between (kβˆ’1+o(1))n(k-1+o(1))n and (k+o(1))n(k + o(1))n. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of (kβˆ’14+o(1))n(k - \frac{1}{4} + o(1))n. We improve the upper bound to (kβˆ’12+o(1))n(k - \frac{1}{2}+ o(1))n. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest

    Ramsey-nice families of graphs

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    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 F∈FF\in\mathcal{F}. We say that F\mathcal{F} is kk-nice if for every graph GG with Ο‡(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 F∈FF\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 kβ‰₯k0(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
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