1,422 research outputs found
The Ramsey number of dense graphs
The Ramsey number r(H) of a graph H is the smallest number n such that, in
any two-colouring of the edges of K_n, there is a monochromatic copy of H. We
study the Ramsey number of graphs H with t vertices and density \r, proving
that r(H) \leq 2^{c \sqrt{\r} \log (2/\r) t}. We also investigate some related
problems, such as the Ramsey number of graphs with t vertices and maximum
degree \r t and the Ramsey number of random graphs in \mathcal{G}(t, \r), that
is, graphs on t vertices where each edge has been chosen independently with
probability \r.Comment: 15 page
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
On path-quasar Ramsey numbers
Let and be two given graphs. The Ramsey number is
the least integer such that for every graph on vertices, either
contains a or contains a . Parsons gave a recursive
formula to determine the values of , where is a path on
vertices and is a star on vertices. In this note, we first
give an explicit formula for the path-star Ramsey numbers. Secondly, we study
the Ramsey numbers , where is a linear forest on
vertices. We determine the exact values of for the cases
and , and for the case that has no odd component.
Moreover, we give a lower bound and an upper bound for the case and has at least one odd component.Comment: 7 page
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