The Minimal Distance Problem

How well separated can a family of point-line pairs in the unit square be? For every $\varepsilon > 0$ there are arbitrarily large families $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$ with $x_i \in \ell_i$ and $\mathrm{dist}(x_i,\ell_j) \ge n^{-2/3-\varepsilon}$ for all $i \ne j$ . Combined with earlier work of Cohen, Pohoata and Zakharov, this settles the problem at the sharp exponent $2/3$ . The same construction disproves a conjecture of Hunter, Pohoata, Verstraete and Zhang about induced point-line matchings over finite fields.

Result: Proved(see note)
Status: Resolved
AI contribution: AI co-developed
Method: Construction
Field: Discrete geometry
Posed by: Cohen, Pohoata, Zakharov
Year posed: 2025
Years open: 1y
Solved: 2026-07-22
Model: GPT-5.6 Pro
Vendor: OpenAI
Collaborators: Cosmin Pohoata
Verification: Unreviewed
Publication: Preprint
Significance: 20 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

also disproves a separate finite-field conjecture of Hunter, Pohoata, Verstraete and Zhang

What the AI did

The acknowledgement draws the line precisely. The author's own plan was to use a high-degree number field analogue of the Hunter-Pohoata-Verstraete-Zhang construction to reach the Ruzsa endpoint. In his words, the decisive new idea of using the codimension-one, square-difference-free, trace-zero lattice in place of a Ruzsa-like set, which is what upgrades the exponent to the sharp one, is entirely due to GPT-5.6 Pro.

Verification

Single-author arXiv preprint; not yet peer-reviewed.

Sources

PaperarXiv:2607.20422 - The sharp exponent for the minimal distance problem
Problem recordCohen et al 2025: Problem statement

The Minimal Distance Problem

What was actually shown

What the AI did

Verification

Sources

Discussion