Unique Sequences Containing No k-Term Arithmetic Progressions

In this paper, we are concerned with calculating r(k, n), the length of the longest k-Ap free subsequences in 1,2,...,n. We prove the basic inequality r(k,n) <= n - [m/2], where n = m(k - 1) + r and r < k - 1. We also discuss a generalization of a famous conjecture of Szekeres (as appears in Erdos and Turan [4]) and describe a simple greedy algorithm that appears to give an optimal k-AP free sequence infinitely often. We provide many exact values of r(k, n) in the Appendix.