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Statement

Let

u(n)

denote the maximum number of unordered pairs at distance exactly

1

determined by

n

points in the Euclidean plane

\mathbb{R}^2

. Erdős (1946) proved

u(n) \geq n^{1 + c/\log\log n}

via a

\sqrt{n} \times \sqrt{n}

integer grid construction (and the May 2026 disproof of his upper-bound conjecture shows

u(n) \geq n^{1.014\ldots}

for infinitely many

n

). The best known upper bound is

u(n) = O(n^{4/3})

. \textbf{Task:} Improve the upper bound exponent below

4/3

, or otherwise advance the asymptotic understanding of

u(n)

.

Current frontier

Upper bound

u(n) = O(n^{4/3})

due to Spencer-Szemerédi-Trotter (1984), unchanged for 40+ years; lower bound

u(n) \geq n^{1.014\ldots}

for infinitely many

n

via the OpenAI / Sawin disproof of the original Erdős upper-bound conjecture (May 2026).

When this counts as solved

The current upper bound

u(n) = O(n^{4/3})

has stood for 40+ years. Closing this out means either pinning down the asymptotically tight value of

u(n)

with matching bounds, or making a strict advance: a rigorous proof that

u(n) = O(n^{4/3 - \varepsilon})

for some explicit

\varepsilon > 0

, a lower bound strictly better than the

n^{1.014\ldots}

from the 2026 disproof, or a structural impossibility theorem (e.g., a proven barrier that rules out an explicit class of approaches from breaking the

4/3

exponent). A counterexample doesn't apply — this is a bound, not a binary conjecture. Refinements that don't move the exponent don't close the problem.

Classification

quantitative