82 research outputs found
Multicolour Ramsey numbers of odd cycles
We show that for any positive integer there exists an integer and a
-colouring of the edges of with no monochromatic odd cycle of
length less than . This makes progress on a problem of Erd\H{o}s and Graham
and answers a question of Chung. We use these colourings to give new lower
bounds on the -colour Ramsey number of the odd cycle and prove that, for all
odd and all sufficiently large, there exists a constant such that
An improvement on Łuczak's connected matchings method
A connected matching in a graph G is a matching contained in a connected component of G. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic connected matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs. We illustrate the potential of this new reduction by showing how it can be used to determine the 3-colour Ramsey number of long paths, using a simpler argument than the original one by Gyárfás, Ruszinkó, Sárközy, and Szemerédi (2007)
Improved bounds on the multicolor Ramsey numbers of paths and even cycles
We study the multicolor Ramsey numbers for paths and even cycles,
and , which are the smallest integers such that every coloring of
the complete graph has a monochromatic copy of or
respectively. For a long time, has only been known to lie between
and . A recent breakthrough by S\'ark\"ozy and later
improvement by Davies, Jenssen and Roberts give an upper bound of . We improve the upper bound to . Our approach uses structural insights in connected graphs without a
large matching. These insights may be of independent interest
Ramsey properties of randomly perturbed graphs: cliques and cycles
Given graphs , a graph is -Ramsey if for every
colouring of the edges of with red and blue, there is a red copy of
or a blue copy of . In this paper we investigate Ramsey questions in the
setting of randomly perturbed graphs: this is a random graph model introduced
by Bohman, Frieze and Martin in which one starts with a dense graph and then
adds a given number of random edges to it. The study of Ramsey properties of
randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in
2006; they determined how many random edges must be added to a dense graph to
ensure the resulting graph is with high probability -Ramsey (for
). They also raised the question of generalising this result to pairs
of graphs other than . We make significant progress on this
question, giving a precise solution in the case when and
where . Although we again show that one requires polynomially fewer
edges than in the purely random graph, our result shows that the problem in
this case is quite different to the -Ramsey question. Moreover, we
give bounds for the corresponding -Ramsey question; together with a
construction of Powierski this resolves the -Ramsey problem.
We also give a precise solution to the analogous question in the case when
both and are cycles. Additionally we consider the
corresponding multicolour problem. Our final result gives another
generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we
determine how many random edges must be added to a dense graph to ensure the
resulting graph is with high probability -Ramsey (for odd
and ).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil
Powierski, stated results for cliques in graphs of low positive density
separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to
appear in CP
3‐Color bipartite Ramsey number of cycles and paths
The k-colour bipartite Ramsey number of a bipartite graph H is the least integer n for which
every k-edge-coloured complete bipartite graph Kn,n contains a monochromatic copy of H. The
study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and,
independently, by Gy´arf´as and Lehel, who determined the 2-colour Ramsey number of paths. In
this paper we determine asymptotically the 3-colour bipartite Ramsey number of paths and (even)
cycles
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