Behrend's Theorem for Dense Subsets of Finite Vector Spaces

Abstract

The “combinatorial line conjecture” states that for all q ≧ 2 and there exists such that if , X is a q-element set, and A is any subset of X n (= cartesian product of n copies of X) with more than elements (that is, A has density greater than ), then A contains a combinatorial line. (For a definition of combinatorial line, together with statements and proofs of many results related to the combinatorial line conjecture, including all those results mentioned below, see [5]. Since we are not directly concerned with combinatorial lines in this paper, we do not reproduce the definition here.)This conjecture (which is a strengthened version of a conjecture of Moser [7]), if true, would bear the same relation to the Hales-Jewett theorem that Szemerédi's theorem bears to van der Waerden's theorem.