201 research outputs found
Integer colorings with forbidden rainbow sums
For a set of positive integers , an -coloring of is
rainbow sum-free if it contains no rainbow Schur triple. In this paper we
initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of
sum-free sets, which asks for the subsets of with the maximum number of
rainbow sum-free -colorings. We show that for , the interval is
optimal, while for , the set is optimal. We
also prove a stability theorem for . The proofs rely on the hypergraph
container method, and some ad-hoc stability analysis.Comment: 20 page
The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques
Let be a sequence of natural numbers. For a
graph , let denote the number of colourings of the edges
of with colours such that, for every , the
edges of colour contain no clique of order . Write
to denote the maximum of over all graphs on vertices.
This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it
has been solved only for a very small number of non-trivial cases.
We prove that, for every and , there is a complete
multipartite graph on vertices with . Also, for every we construct a finite
optimisation problem whose maximum is equal to the limit of as tends to infinity. Our final result is a
stability theorem for complete multipartite graphs , describing the
asymptotic structure of such with in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So
Ordered Ramsey numbers of loose paths and matchings
For a -uniform hypergraph with vertex set , the
ordered Ramsey number is the least integer such
that every -coloring of the edges of the complete -uniform graph on
vertex set contains a monochromatic copy of whose vertices
follow the prescribed order. Due to this added order restriction, the ordered
Ramsey numbers can be much larger than the usual graph Ramsey numbers. We
determine that the ordered Ramsey numbers of loose paths under a monotone order
grows as a tower of height one less than the maximum degree. We also extend
theorems of Conlon, Fox, Lee, and Sudakov [Ordered Ramsey numbers,
arXiv:1410.5292] on the ordered Ramsey numbers of 2-uniform matchings to
provide upper bounds on the ordered Ramsey number of -uniform matchings
under certain orderings.Comment: 13 page
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