Unique Sequences Containing No kkk-Term Arithmetic Progressions (original) (raw)

Abstract

In this paper, we are concerned with calculating r(k,n)r(k, n)r(k,n), the length of the longest kkk-AP free subsequences in 1,2,ldots,n1, 2, \ldots , n1,2,ldots,n. We prove the basic inequality r(k,n)len−lfloorm/2rfloorr(k, n) \le n − \lfloor m/2\rfloorr(k,n)len−lfloorm/2rfloor, where n=m(k−1)+rn = m(k − 1) + rn=m(k−1)+r and r<k−1r < k − 1r<k−1. We also discuss a generalization of a famous conjecture of Szekeres (as appears in Erdős and Turán) and describe a simple greedy algorithm that appears to give an optimal kkk-AP free sequence infinitely often. We provide many exact values of r(k,n)r(k, n)r(k,n) in the Appendix.