4 research outputs found
Colouring set families without monochromatic k-chains
A coloured version of classic extremal problems dates back to Erd\H{o}s and
Rothschild, who in 1974 asked which -vertex graph has the maximum number of
2-edge-colourings without monochromatic triangles. They conjectured that the
answer is simply given by the largest triangle-free graph. Since then, this new
class of coloured extremal problems has been extensively studied by various
researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of
Sperner's Theorem, the classic result in extremal set theory on the size of the
largest antichain in the Boolean lattice, and Erd\H{o}s' extension to
-chain-free families.
Given a family of subsets of , we define an
-colouring of to be an -colouring of the sets without
any monochromatic -chains . We
prove that for sufficiently large in terms of , the largest
-chain-free families also maximise the number of -colourings. We also
show that the middle level, , maximises the
number of -colourings, and give asymptotic results on the maximum
possible number of -colourings whenever is divisible by three.Comment: 30 pages, final versio
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