14,009 research outputs found
New Computational Upper Bounds for Ramsey Numbers R(3,k)
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
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
For graphs and , we say is Ramsey for if every -coloring of
the edges of contains a monochromatic copy of . The graph is Ramsey
-minimal if is Ramsey for and there is no proper subgraph of
so that is Ramsey for . Burr, Erdos, and Lovasz defined to
be the minimum degree of over all Ramsey -minimal graphs . Define
to be a graph on vertices consisting of a complete graph on
vertices and one additional vertex of degree . We show that
for all values ; it was previously known that , so it
is surprising that is much smaller.
We also make some further progress on some sparser graphs. Fox and Lin
observed that for all graphs , where is
the minimum degree of ; Szabo, Zumstein, and Zurcher investigated which
graphs have this property and conjectured that all bipartite graphs without
isolated vertices satisfy . Fox, Grinshpun, Liebenau,
Person, and Szabo further conjectured that all triangle-free graphs without
isolated vertices satisfy this property. We show that -regular -connected
triangle-free graphs , with one extra technical constraint, satisfy ; the extra constraint is that has a vertex so that if one
removes and its neighborhood from , the remainder is connected.Comment: 10 pages; 3 figure
Ramsey Goodness and Beyond
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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