When $d=2$ the answer is $6$ (due to Kelly [ErKe47] - an alternative proof is given by Kovács [Ko24c]). When $d=3$ the answer is $8$ (due to Croft [Cr62]). The best upper bound known in general is due to Blokhuis [Bl84] who showed that\[\lvert A\rvert \leq \binom{d+2}{2}.\]Alweiss has observed a lower bound of $\binom{d+1}{2}$ follows from considering the subset of $\mathbb{R}^{d+1}$ formed of all vectors $e_i+e_j$ where $e_i,e_j$ are distinct coordinate vectors. This set can be viewed as a subset of some $\mathbb{R}^d$, and is easily checked to have the required property.

Weisenberg observed in the comments that an additional point can be added to Alweiss' construction, giving a lower bound of $\binom{d+1}{2}+1$.

The fact that the truth for $d=3$ is $8$ suggests that neither of these bounds is the truth.

See also [1088] for a generalisation.

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This page was last edited 28 October 2025.