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on Erdős unit distance exponent ($C_{84a}$)

The statement

For a finite set $P \subset \mathbb{R}^2$ of $n$ points, let

u(P) := \lvert \{ \{x,y\} \subset P : \lvert x-y \rvert = 1 \} \rvert

denote the number of unordered pairs of points in $P$ at unit distance, and let $u(n) := \max_{\lvert P\rvert = n} u(P)$ denote the maximum number of unit distances among any set of $n$ points in the plane. The Erdős unit distance exponent is

C_{84} := \limsup_{n \to \infty} \frac{\log u(n)}{\log n}.

Equivalently, $C_{84}$ is the infimum of all $\alpha$ for which $u(n) \ll_\alpha n^\alpha$ as $n \to \infty$ .

The problem was introduced by Erdős in 1946 [E1946], who conjectured that $u(n) = n^{1+o(1)}$ , i.e. $C_{84} = 1$ . This conjecture was disproved in May 2026 by an OpenAI internal model [O2026], with a human-verified writeup given in [ABGLSSTWW2026]; an explicit improved lower bound was obtained shortly afterwards by Sawin [S2026].

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.