Given positive integers m and k, a k-term semi-progression of scope m is a sequence x1, x2, ..., xk such that xj+1 − xj ∈ {d, 2d, . . . ,md}, 1 ≤ j ≤ k − 1, for some positive integer d. Thus an arithmetic progression is a semi-progression of scope 1. Let Sm(k) denote the least integer for which every 2-coloring of {1, 2, ..., Sm(k)} yields a monochromatic k-term semi-progression of scope m. We ...

It is shown that for each integer k > 3, there exists a set Sk of positive integers containing no arithmetic progression of k terms, such that 2„6Si \/n > (1 e)k log A:, with a finite number of exceptional k for each real e > 0. This result is shown to be superior to that attainable with other sets in the literature, in particular Rankin's sets &(k), which have the highest known asymptotic dens...

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 ari...