The set of all possible numbers of unit distances for a configuration of $n$ points.

This lemma confirms that the set of possible unit distance counts is bounded above, which ensures that taking the supremum (sSup) is a well-defined operation. The trivial upper bound is the total number of pairs of points, $\binom{n}{2}$.

The maximum number of unit distances determined by any set of $n$ points in the plane. This function is often denoted as $u(n)$ in combinatorics.

Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(\frac{1}{\log\log n})}$ many pairs which are distance $1$ apart?

This was disproved by an internal model at OpenAI, which constructed (for infinitely many $n$) a set $P$ of $n$ points in $\mathbb{R}^2$ such that the number of unit distance pairs in $P$ is at least $n^{1+c}$, where $c > 0$ is an absolute constant.

Constructive form of the disproof. There is an absolute constant $c > 0$ such that infinitely many $n$ admit a configuration realising at least $n^{1+c}$ unit distances.

This is the qualitative content of Theorem 1.1 of Alon–Bloom–Gowers–Litt–Sawin–Shankar– Tsimerman–Wang–Matchett Wood, Remarks on the disproof of the unit distance conjecture (2026). An explicit bound $c \ge 0.014114$ is given by Sawin, An explicit lower bound for the unit distance problem (2026); see erdos_90.variants.sawin_explicit below.

Sawin's explicit exponent. The constructive disproof can be realised with $c \ge 0.014114$ (absorbing the implicit constant $C$ of Sawin's Theorem 1 into a slightly smaller exponent for all large enough $n$). Reference: Theorem 1 of Sawin, arXiv:2605.20579 (2026).

Sawin's explicit bound implies the qualitative polynomial lower bound, by taking $c = 0.014114$.

The polynomial lower bound implies the answer to Erdős 90 is False: a fixed positive exponent $c$ is incompatible with the conjectured $O(1 / \log \log n)$ growth.

Sawin's Lemma 2: lattice geometry of unit distances (Sawin, arXiv:2605.20579).

Let $d \ge 1$, $R \ge 2$, and suppose $\Lambda \subset \mathbb{R}^{2d}$ is a lattice equipped with an additive embedding $\pi : \Lambda \to \mathbb{R}^2$. Suppose $S \subseteq \Lambda$ is a finite set of "matching" vectors satisfying $|v| \le 1$ and $|\pi v| = 1$ for every $v \in S$. Then there is a finite point set $U \subset \mathbb{R}^2$ with unit-distance density at least $(1 - 1/R)^{2d},#S$, i.e. $(1-1/R)^{2d},#S,#U \le #{\text{unit pairs in } U}$.

This pure geometry-of-numbers reduction is the elementary heart of the disproof.

Sawin's Lemmas 11–12 / Remarks Proposition 2.3: the totally real tower.

There exist $rdBound : \mathbb{R}$ and a single infinite set $Q$ of rational primes $q \equiv 1 \pmod 4$ such that for every $N$ one can find a totally real number field $F/\mathbb{Q}$ of degree $\ge N$ with bounded root discriminant $|disc F|^{1/[F:\mathbb{Q}]} \le rdBound$ in which every prime $q \in Q$ splits completely.

The load-bearing feature is the quantifier order: $Q$ is fixed before $F$, so the same primes split completely in fields of unbounded degree. (For a single fixed $F$, Chebotarev already gives infinitely many completely split primes $\equiv 1 \pmod 4$, so a per-field statement would be vacuous.) This uniform splitting in an unbounded tower is the key arithmetic input to the disproof. It is proved as Lemmas 11–12 of Sawin, arXiv:2605.20579, and as Proposition 2.3 of the Remarks paper, via the Golod–Shafarevich inequality for pro-$2$ groups together with the Hajir–Maire–Ramakrishna (2003) tower construction.

A "completely split" rational prime $q$ in $F$ is one for which $(q)$ is the product of exactly $[F:\mathbb{Q}]$ distinct maximal ideals.