Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Claims
2018_02_12_glock_kuhn_lo_osthus: For every fixed k there are partial Steiner triple systems on n vertices with (1/6-o(1))n^2 triples and no j vertices spanning j-2 triples for any j from 4 to k, which with linearity settles the corrected Statement; refereed in Combinatorica and credited by the site's curator.
2018_08_03_bohman_warnke: For every fixed girth bound there are partial Steiner triple systems on n vertices with (1-n^(-beta))n^2/6 triples and no j vertices spanning j-2 triples for j up to the bound, which with linearity settles the corrected Statement; refereed and credited by the site's curator.
2018_09_06_glock: Correct, but answers the site's wording (the single family F_5), not the corrected Statement (the cumulative family), so it does not count toward the problem's standing. The most triples on n points with no three spanning at most five points is (1+o(1))n^2/5, refereed in Bull. Lond. Math. Soc.
2022_09_28_glock_joos_kim_kuhn_lichev_pikhurko: Correct, but answers the site's wording (the single family F_6), not the corrected Statement (the cumulative family), so it does not count toward the problem's standing. The (6,4) limit is 7/36, not 1/6, refereed in Proc. Amer. Math. Soc. Ser. B.
2024_03_07_glock_kim_lichev_pikhurko_sun: Correct, but answers the site's wording (the single families F_7, F_8 and F_9), not the corrected Statement (the cumulative family), so it does not count toward the problem's standing. The (7,5), (8,6) and (9,7) limits are 1/5, 61/330 and 1/5, not 1/6, refereed in Canad. J. Math.
2025_06_02_pikhurko_sun: Correct, but answers the site's wording (the single family F_10), not the corrected Statement (the cumulative family), so it does not count toward the problem's standing. The (10,8) limit is at least 3/16, above 1/6, refereed in European J. Combin.
2026_08_17_alexeev: Answers the site's wording (the single family F_k), not the corrected Statement (the cumulative family), so it does not count toward the problem's standing. A Lean file in Boris Alexeev's lean-proofs collection, written by Codex and GPT-5.6 Sol, builds (5,3)-free triple systems of density 92/529 > 1/6; not built here.