On $k$-free sequences of integers

Abstract:

Let ${A^{(k)}}(n)$ denote the cardinality of the largest subsequence of $0,1,2, \cdots ,n - 1$, which contains no $k$ numbers in arithmetical progression. (Such a sequence is called $k$-free.) ${A^{(k)}}(n)$ is computed (on an IBM 360/65) for $3 \leqq k \leqq 8$, and various values of $n$ to about 50. The results support the old conjecture that for all $k$, the limit ${\tau ^{(k)}} = {\lim _{n \to \infty }}({A^{(k)}}(n))/n = 0$. The results ${\tau ^{(5)}} < .649,{\tau ^{(6)}} < .721,{\tau ^{(7)}} < .776$, and ${\tau ^{(8)}} < .8071$ are obtained. Several cases of a (disproved) conjecture of G. Szekeres are verified, including ${A^{(5)}}(94) = 64$.