Sequences Containing No 3-Term Arithmetic Progressions

Abstract

A subsequence of the sequence $(1,2,...,n)$ is called a 3-$AP$-free sequence if it does not contain any three term arithmetic progression. By $r(n)$ we denote the length of the longest such 3-$AP$-free sequence. The exact values of the function $r(n)$ were known, for $n\leq 27$ and $41\leq n \leq 43$. In the present paper we determine, with a use of computer, the exact values, for all $n\leq 123$. The value $r(122)=32$ shows that the Szekeres' conjecture holds for $k=5$.