14,009 research outputs found

    New Computational Upper Bounds for Ramsey Numbers R(3,k)

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    Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <= 68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are improvements by one over the previously best known bounds. Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on n vertices without independent sets of order k. The new upper bounds on R(3,k) are obtained by completing the computation of the exact values of e(3,k,n) for all n with k <= 9 and for all n <= 33 for k = 10, and by establishing new lower bounds on e(3,k,n) for most of the open cases for 10 <= k <= 15. The enumeration of all graphs witnessing the values of e(3,k,n) is completed for all cases with k <= 9. We prove that the known critical graph for R(3,9) on 35 vertices is unique up to isomorphism. For the case of R(3,10), first we establish that R(3,10) = 43 if and only if e(3,10,42) = 189, or equivalently, that if R(3,10) = 43 then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that R(3,10) <= 42.Comment: 28 pages (includes a lot of tables); added improved lower bound for R(3,11); added some note

    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

    Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques

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    For graphs FF and HH, we say FF is Ramsey for HH if every 22-coloring of the edges of FF contains a monochromatic copy of HH. The graph FF is Ramsey HH-minimal if FF is Ramsey for HH and there is no proper subgraph F′F' of FF so that F′F' is Ramsey for HH. Burr, Erdos, and Lovasz defined s(H)s(H) to be the minimum degree of FF over all Ramsey HH-minimal graphs FF. Define Ht,dH_{t,d} to be a graph on t+1t+1 vertices consisting of a complete graph on tt vertices and one additional vertex of degree dd. We show that s(Ht,d)=d2s(H_{t,d})=d^2 for all values 1<d≤t1<d\le t; it was previously known that s(Ht,1)=t−1s(H_{t,1})=t-1, so it is surprising that s(Ht,2)=4s(H_{t,2})=4 is much smaller. We also make some further progress on some sparser graphs. Fox and Lin observed that s(H)≥2δ(H)−1s(H)\ge 2\delta(H)-1 for all graphs HH, where δ(H)\delta(H) is the minimum degree of HH; Szabo, Zumstein, and Zurcher investigated which graphs have this property and conjectured that all bipartite graphs HH without isolated vertices satisfy s(H)=2δ(H)−1s(H)=2\delta(H)-1. Fox, Grinshpun, Liebenau, Person, and Szabo further conjectured that all triangle-free graphs without isolated vertices satisfy this property. We show that dd-regular 33-connected triangle-free graphs HH, with one extra technical constraint, satisfy s(H)=2δ(H)−1s(H) = 2\delta(H)-1; the extra constraint is that HH has a vertex vv so that if one removes vv and its neighborhood from HH, the remainder is connected.Comment: 10 pages; 3 figure

    Ramsey Goodness and Beyond

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    In a seminal paper from 1983, Burr and Erdos started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving all but one of the Burr-Erdos problems.Comment: A new reference is adde
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