Optimal off-diagonal separation in incident point–line configurations
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Choose n incident point–line pairs inside the unit square. How large can the minimum distance from each point to every line other than its own be? The target exponent is n^(−2/3+o(1)).
Why it matters and what counts as progress
Why it matters: Matching exponents would close the gap between incidence-based upper bounds and explicit separated configurations.
What counts as progress: Constructions with off-diagonal separation at least n^(−2/3−ε) for every ε>0, or a sharper general upper bound.
How the problem developed
- 11 Sept 2024 · Upper bound [1]
Alex Cohen, Cosmin Pohoata and Dmitrii Zakharov Introduced the asymptotic minimal point-line separation problem and proved an upper bound n^(−2/3+o(1)).
- 22 Jul 2026 · Matching exponent [2]
Cosmin Pohoata Constructed arbitrarily large incident point-line configurations with off-diagonal separation at least n^(−2/3−ε) for each fixed ε>0, matching the earlier exponent.
AI contribution
The sharp exponent for the minimal distance problem
Pohoata constructs n incident point-line pairs in [0,1]² whose off-diagonal distances are at least n^(−2/3−ε) for every ε>0 and all sufficiently large n. With the previous upper bound, the maximum separation is n^(−2/3+o(1)).
Sources
Website timeline
Publication dates and editorial changes.
GPT-origin construction gives the minimal-distance sharp exponent