A generalization of sets without long arithmetic progressions based on Szekeres algorithm

George Szekeres described some subsets of {1,…,n} without arithmetic progressions of length p for odd primes p, obtained by a greedy algorithm. Let rk(n) denote the size of the largest subset of {1,…,n} without arithmetic progressions of length k. In this paper, the history of results based on the constructions by Szekeres is briefly surveyed. New inequalities for rk(n) and van der Waerden numbers are derived by generalizing these constructions. In particular, for any odd prime p, we prove that rp(p2)⩾(p−1)2+tp, where limp→∞tplnp=1.