1,565 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 largest -sum-free subsets
Let be the infimum of the largest sum-free subset of
any set of positive integers. An old conjecture in additive combinatorics
asserts that there is a constant and a function
as , such that . The
constant is determined by Eberhard, Green, and Manners, while the
existence of is still wide open.
In this paper, we study the analogous conjecture on -sum-free sets
and restricted -sum-free sets. We determine the constant
for every -sum-free sets, and confirm the conjecture for infinitely
many .Comment: 33 pages; accepted for publication in Trans. Amer. Math. So
- β¦