74 research outputs found
Rainbow-free 3-colorings in abelian groups
A 3–coloring of an abelian group G is rainbow–free if there is no 3–term arithmetic
progression with its members having pairwise distinct colors. We describe
the structure of rainbow–free colorings of abelian groups. This structural description
proves a conjecture of Jungi´c et al. on the size of the smallest chromatic class of a rainbow–free coloring of cyclic groups.Postprint (published version
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
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