Problem detail · source-aware
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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.
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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.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Single-author arXiv preprint; not yet peer-reviewed.
Correctness: unknown · statement fidelity: unaudited · peer review: none
also disproves a separate finite-field conjecture of Hunter, Pohoata, Verstraete and Zhang
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.