Claims

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1981_12_01_komlos_pintz_szemeredi: Komlós, Pintz and Szemerédi prove that any n points in the unit square span a triangle of area at most exp(c sqrt(log n)) n^(-8/7), the exponent 8/7 for the disk, since superseded by Cohen, Pohoata and Zakharov; refereed.

1982_02_01_komlos_pintz_szemeredi: Komlós, Pintz and Szemerédi prove that for all large n some n points of the unit square have every triangle of area at least c (log n)/n^2, which gives the lower bound alpha(n) >> (log n)/n^2 for the disk; refereed.

2017_03_08_ellmann: An arXiv preprint placing n points in the unit circle with every triangle of area at least of order n^(-3/2) (log n)^(-7/2); its latest version calls the method heuristic and rests on an unproved uniformity assumption.

2020_06_05_agama: An arXiv preprint asserting both a lower bound of order (log n)/n^(3/2) and an upper bound n^(-3/2+epsilon) for the disk, so an estimate of alpha(n); rejected on the release's record of a collinear triple in its lower bound.

2024_09_11_cohen_pohoata_zakharov: Cohen, Pohoata and Zakharov prove that any n points in the unit square span a triangle of area at most n^(-7/6+o(1)), which gives the upper bound alpha(n) << n^(-7/6+o(1)) for the disk; refereed in Inventiones.

2026_09_25_openai: For every large n there are n points in the unit square with every triangle of area at least c n^(-2+eta) for an absolute, extremely small eta, which refutes the almost-n^(-2) formulation; the bound carries over to the disk.

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