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On some sequences of integers. (English) Zbl 0015.15203
Let \(a_1 < a_2 < \cdots < a_r \leq n\) be a set of positive integers such that \(a_i-a_j\neq a_j-a_k\) for \(1\leq k < j < i\leq r\). For given \(n\) let \(r(n)\) be the maximum value of \(r\) for which such a set exists. The authors prove that (1) \(r(2n)\leq n\) for \(n\geq 8\), (2) \(\limsup \frac{r(n)}{n} \leq \frac 49\). They conjecture that \(r(n)=o(n)\), and G. Szekeres conjectured that \(r(\tfrac12 (3^k+1)) =2^k\).

MSC:
11B75 Other combinatorial number theory
11B05 Density, gaps, topology
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