652 research outputs found
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
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
Induced Ramsey-type theorems
We present a unified approach to proving Ramsey-type theorems for graphs with
a forbidden induced subgraph which can be used to extend and improve the
earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham,
and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by
Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's
regularity lemma, thereby giving much better bounds. The same approach can be
also used to show that pseudo-random graphs have strong induced Ramsey
properties. This leads to explicit constructions for upper bounds on various
induced Ramsey numbers.Comment: 30 page
- …