files / problems / caccetta-haggkvist-k3 / compute / q1 / certs / keep / n18_summary.json

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{
  "n": 18,
  "d": 6,
  "published_hkn": 0.3465,
  "repo_f4": 0.34645,
  "personal_communication_3388": 0.3388,
  "why_open": "ceil(18/3)=6 < 0.3465*18=6.237. First exact order not implied by Hoang-Reed or HKN.",
  "encoding": "Sinz sequential counters, N+(0)={1..6}, cubes on k=|N-(0)| with N-(0)={7..6+k}",
  "cubes": {
    "k11": "UNSAT DRAT stored",
    "k10": "UNSAT DRAT stored",
    "k9": "UNSAT DRAT stored",
    "k8": "UNSAT DRAT stored",
    "k7": "UNSAT DRAT stored",
    "k6": "UNSAT DRAT stored",
    "k5": "UNSAT DRAT verified 2.6MB, regenerated not stored",
    "k4": "UNSAT DRAT verified 14MB, regenerated not stored",
    "k3": "UNSAT DRAT verified 52MB, regenerated not stored",
    "k2": "UNSAT DRAT verified 245MB, regenerated not stored",
    "k1": "UNSAT 587s, DRAT VERIFIED 1207902119 bytes, sha256 73441d6e484a246f31c22e3a636ab764aef2624a52aa7dbe94f3d3d04bdcbf73, not stored",
    "k0": "UNKNOWN at 600s; t=1..5 UNSAT DRAT stored; t=6 UNKNOWN at 900s stable"
  },
  "pigeonhole": "6-outregular on 18 vertices has 108 arcs, so some in-degree is at least 6. Exact CH at n=18 reduces to cubes k=6..11.",
  "exact_n18": "UNSAT for k=6..11 with stored DRAT. Every 18-vertex oriented graph with min out-degree 6 has a directed triangle.",
  "unused_search": "k=0 t=6 still UNKNOWN; not needed once the pigeonhole is used"
}