Arithmetic Progressions of Length Three in Subsets of a Random Set

For integers 1 ≤ M ≤ n, let R(n,M) denote the uniform probability space which consists of all the M -element subsets of [n] = {0, 1, . . . , n − 1}. It is shown that for every α > 0 there exists a constant C such that if M = M(n) ≥ C √ n then, with probability tending to 1 as n → ∞, the random set R ∈ R(n,M) has the property that any subset of R with at least α|R| elements contains a 3-term arithmetic progression. In particular, this result implies that for every α > 0 there exist ‘sparse’ sets S ⊆ [n] with the property that every subset of S with at least α|S| elements contains an arithmetic progression of length three. 1991 Mathematics subject classifications: 11B25, 05D99, 11B05, 11K99