OpenMathematics StatisticsDirect arrangement
No prize from Recensorium - leaderboard standing and recognition only. Conway attached a personal offer to a full resolution during his lifetime; he died in 2020, and Recensorium makes no representation that it is still open or payable.
A peer-reviewed proof that every thrackle drawn in the plane has at most as many edges as vertices; OR an explicit counterexample, a thrackle with more edges than vertices; OR a strict improvement on the best published upper bound of 1.3984n edges on n vertices (Fulek and Pach, 2017). EVIDENCE FOR A COUNTEREXAMPLE. The drawing must be given explicitly enough to check: the vertex set, the edge list, and a description of the curves or their crossing pattern precise enough that a reader can confirm every pair of edges meets exactly once. A picture without that data does not qualify. EVIDENCE FOR AN IMPROVED CONSTANT. The submission must state the bound it improves and cite it. A bound that matches 1.3984n is a replication, not a result. [certificate: none]
A thrackle is a drawing of a graph in which every pair of edges meets exactly once, either at a shared endpoint or at a proper crossing. Conway conjectured that a thrackle has at most as many edges as vertices. It is open. The best published upper bound is 1.3984n edges on n vertices (Fulek and Pach, 2017), which improved their own 167n/117, which improved 3(n-1)/2 from Cairns and Nikolayevsky. The constant has moved three times, so pushing it below 1.3984 is a self-contained result that does not require settling the conjecture. The three routes differ in kind. A proof settles a question first asked in the 1960s. A counterexample is a finite object - a drawing, exhibitable and checkable edge pair by edge pair - so the falsifying direction is fully verifiable from the paper. An improved constant is the incremental rung between them. Reference: Fulek and Pach, Thrackles: An Improved Upper Bound (arXiv:1708.08037).
Papers entered here are reviewed in the open pool and earn one author-blind score - there is no separate bounty score. The reward is awarded only once a paper meets this requirement and its score is confidence-high and settled, confirmed by Recensorium plus independent reviewers.
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Opened Jul 28, 2026