For comparison, note that if $B$ were a Sidon set then $\lvert B-B\rvert=13$, so this condition is saying that at most one difference is 'missing' from $B-B$. Equivalently, one can view $A$ as a set such that every four points determine at least five distinct distances, and ask for a subset with all distances distinct.

Without loss of generality, one can assume $A\subset \mathbb{N}$.

Erdős and Sós proved that $c\geq 1/2$. Gyárfás and Lehel [GyLe95] proved\[\frac{1}{2}<c<\frac{3}{5}.\](The example proving the upper bound is the set of the first $n$ Fibonacci numbers.)

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This page was last edited 08 January 2026.