{"id":"openai-math/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders","pdf":"https://github.com/openai/math/blob/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026","tags":["hypergraphs","combinatorics","covering-number"],"score":0,"title":"Balanced counterexamples to Ryser's conjecture at prime orders","domain":"Combinatorics","family":"162","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"For every sufficiently large prime q , we construct a finite intersecting $`(q+1)`$-partite $`(q+1)`$-uniform hypergraph with covering number $`q+1`$ and exactly $`q+1`$ nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-27","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=to%20meet%20every%20edge%2C%20whereas%20the%20conjecture%20predicts","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"to meet every edge, whereas the conjecture predicts . Crossing numbers of complete and complete bipartite graphs. These problems appeared in my discussion of Turán’s mathematics . The release reports proofs of the Harary–Hill and Zarankiewicz formulas, determining the minimum number of crossings…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"}],"formalized":false,"family_title":"Counterexamples to Ryser’s covering conjecture"}