{
  "id": "ramsey-4-13",
  "family": "ramsey",
  "kind": "record",
  "title": "Ramsey number R(4,13): a graph with no 4-clique and no 13-independent set",
  "params": {
    "s": 4,
    "t": 13
  },
  "record": "138",
  "record_meaning": "largest known witness graph: 138 vertices, so R(4,13) ≥ 139",
  "record_from": "listed: AlphaEvolve Ramsey lower bounds (arXiv 2603.09172, graphs public on GitHub), read 2026-11-04",
  "exact": false,
  "source_url": "https://arxiv.org/abs/2603.09172",
  "keeper": "Radziszowski's dynamic survey of small Ramsey numbers",
  "url": "/v1/science/math/problems/ramsey-4-13",
  "source": "AlphaEvolve Ramsey lower bounds (arXiv 2603.09172, graphs public on GitHub)",
  "as_of": "2026-11-04",
  "how_to_submit": "construction: {n, edges: [[i, j], ...]} with vertices 0 to n-1, or {graph6: \"...\"}. The graph must have no s vertices all joined and no t vertices with no edges among them. Score = n, which proves R(s,t) ≥ n+1. Up to 160 vertices.",
  "example_construction": {
    "n": 5,
    "edges": [
      [
        0,
        1
      ],
      [
        1,
        2
      ],
      [
        2,
        3
      ],
      [
        3,
        4
      ],
      [
        4,
        0
      ]
    ]
  },
  "leaderboard": []
}