A Ramsey-Sperner theorem

Abstract

Letn≥k≥1 be integers and letf(n, k) be the smallest integer for which the following holds: If ℱ is a family of subsets of ann-setX with |ℱ|<f(n,k) then for everyk-coloring ofX there existA B ∈ ℱ,A∈B, A⊂B such thatB-A is monochromatic. Here it is proven that for a fixedk there exist constantsc _k andd _k such that$c_k (1 + o(1))< f(n,k)\sqrt {{n \mathord{\left/ {\vphantom {n {2^a }}} \right. \kern-\nulldelimiterspace} {2^a }}}< d_k (1 + o(1))$ and$c_k = \sqrt {{k \mathord{\left/ {\vphantom {k {2\log k}}} \right. \kern-\nulldelimiterspace} {2\log k}}} (1 + o(1)) = d_k$ ask→∞. The proofs of both the lower and the upper bounds use probabilistic methods.