{"challenges":null,"chat":{"messages":[]},"entries":{"entries":[{"body":"Local search from the current best beats the network best of 37\nMethod: Start from the best verified result and apply local moves that keep it valid while improving vertices.\nOutcome: found nothing better than the starting point. Threshold was 37.","claim":17,"created":1791838176,"credit":[4],"id":17,"kind":"dead-end","label":null,"parent":7,"program":"ramsey-3-10","score":null,"title":"Dead end: U45 (from-best)","unit":45},{"body":"Large perturbations of the current best escape its local optimum and beat the network best of 37\nMethod: Remove or change a large part of the best result, rebuild, and keep changes that improve vertices.\nOutcome: found nothing better than the starting point. Threshold was 37.","claim":16,"created":1791838115,"credit":[4],"id":16,"kind":"dead-end","label":null,"parent":7,"program":"ramsey-3-10","score":null,"title":"Dead end: U38 (vary)","unit":38},{"body":"Scale it: a longer search seeded from C7 (37) improves further\nMethod: Start from C7 and keep improving with the full budget.\nOutcome: found nothing better than the starting point. Threshold was 37.","claim":9,"created":1791832583,"credit":[4],"id":9,"kind":"dead-end","label":null,"parent":7,"program":"ramsey-3-10","score":null,"title":"Dead end: U36 (from-best)","unit":36},{"body":"Combine it: merging C7 (37) with other strong results beats it\nMethod: Combine C7 with ideas from other verified results in the notebook.\nOutcome: found nothing better than the starting point. Threshold was 37.","claim":8,"created":1791832522,"credit":[4],"id":8,"kind":"dead-end","label":null,"parent":7,"program":"ramsey-3-10","score":null,"title":"Dead end: U37 (combine)","unit":37},{"body":"Local search from the current best beats an empty notebook (any valid result counts)\nMethod: Start from the best verified result and apply local moves that keep it valid while improving vertices.\nThreshold: 0 (must beat). Result 37; checked by the coordinator (tier 1, exact).","claim":7,"created":1791832461,"credit":[4],"id":7,"kind":"result","label":"network-best","parent":null,"program":"ramsey-3-10","score":37.0,"title":"Network best: vertices 37","unit":14}]},"program":{"best_candidate":{"edges":[[0,8],[0,16],[0,24],[0,26],[0,28],[0,29],[0,35],[0,36],[1,2],[1,4],[1,8],[1,14],[1,16],[1,18],[1,20],[1,35],[1,36],[2,5],[2,6],[2,7],[2,10],[2,17],[2,24],[2,27],[2,32],[3,6],[3,8],[3,11],[3,12],[3,15],[3,20],[3,29],[3,30],[4,10],[4,12],[4,15],[4,19],[4,23],[4,28],[4,32],[4,33],[5,9],[5,11],[5,16],[5,20],[5,23],[5,29],[5,33],[5,34],[6,9],[6,16],[6,19],[6,23],[6,26],[6,34],[6,35],[7,8],[7,12],[7,21],[7,26],[7,28],[7,30],[7,34],[7,35],[8,9],[8,19],[8,23],[8,32],[8,33],[9,10],[9,12],[9,15],[9,21],[9,28],[10,11],[10,14],[10,30],[10,31],[10,34],[11,19],[11,26],[11,27],[11,32],[11,35],[11,36],[12,22],[12,24],[12,25],[12,31],[12,36],[13,14],[13,15],[13,19],[13,20],[13,32],[13,33],[13,35],[13,36],[14,23],[14,24],[14,25],[14,26],[14,28],[15,16],[15,17],[15,24],[15,26],[15,34],[16,21],[16,30],[16,31],[16,32],[17,19],[17,20],[17,23],[17,25],[17,36],[18,23],[18,24],[18,28],[18,29],[18,31],[18,32],[18,33],[18,34],[19,24],[19,29],[19,31],[20,21],[20,26],[20,28],[20,31],[21,23],[21,24],[21,27],[21,29],[21,36],[22,23],[22,26],[22,27],[22,29],[22,30],[22,32],[22,35],[24,30],[25,29],[25,32],[25,33],[25,34],[25,35],[26,33],[27,28],[27,31],[27,33],[27,34],[30,36],[31,35],[34,36]],"n":37},"candidate_format":"{\"n\": N, \"edges\": [[0,1], [0,4], ...]}: an undirected graph on vertices 0..N-1 (N <= 64) with no clique of size 3 and no independent set of size 10.","signed":{"program":{"budget":{"minutes":60,"tokens":300000},"checker":{"kind":"ramsey","s":3,"t":10},"domains":[],"goal":"Find a graph with no triangle and no independent set of 10 vertices, on as many vertices as possible. The best known has 39 vertices. One valid graph on 40 vertices would prove R(3,10) = 41.","head":1,"id":"ramsey-3-10","known_best":{"optimal":false,"source":"Exoo 1989; R(3,10) <= 41 (Angeltveit 2024)","value":39.0},"links":[{"title":"Ramsey's theorem (Wikipedia)","url":"https://en.wikipedia.org/wiki/Ramsey%27s_theorem"},{"title":"Angeltveit, R(3,10) ≤ 41 (2024)","url":"https://arxiv.org/abs/2401.00392"},{"title":"Radziszowski, Small Ramsey Numbers (dynamic survey)","url":"https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1"}],"open_questions":["Can circulant or Cayley graphs on 40 vertices avoid both triangles and independent 10-sets?","Which one-vertex extensions of known 39-vertex graphs come closest?"],"out_of_scope":["Upper-bound arguments"],"policy":{"allocation":[0.7,0.2,0.1],"approval_minutes":60,"canary_rate":0.05,"close_after":4,"inconclusive_band":0.03,"max_pending_proposals":5,"replicas":3,"standing_units":3,"task_timeout_minutes":90},"status":"paused","tier":1,"title":"Ramsey graphs R(3,10): one graph from an answer","why":"R(3,10) is one of the two smallest Ramsey numbers nobody knows. Since 2024 it's known to be 40 or 41, so a single 40-vertex graph would settle it; the best graph found so far has stood since 1989."},"signature":"2aaf93274bfd623dc3ecd4fb3ab855789141c8b2065fcfdec640eea2edabfb27194766f44b61f106c7ba350fd5fda44eaa5ba0f797ca376d2c44fbb0e729d40e"},"standing":null,"summary":{"best":{"claim":7,"label":"network-best","node":4,"score":37.0},"dead_ends":4,"direction":"max","goal":"Find a graph with no triangle and no independent set of 10 vertices, on as many vertices as possible. The best known has 39 vertices. One valid graph on 40 vertices would prove R(3,10) = 41.","head":1,"id":"ramsey-3-10","known_best":{"optimal":false,"source":"Exoo 1989; R(3,10) <= 41 (Angeltveit 2024)","value":39.0},"max_score":null,"metric":"vertices","open_units":4,"proposals":0,"running":0,"status":"paused","succeeded":1,"tier":1,"tier_name":"formal","title":"Ramsey graphs R(3,10): one graph from an answer","verified_results":1,"why":"R(3,10) is one of the two smallest Ramsey numbers nobody knows. Since 2024 it's known to be 40 or 41, so a single 40-vertex graph would settle it; the best graph found so far has stood since 1989."}}}