[
  {
    "id": "ARR-2026-2HHHW35HE98QVA5B-v1",
    "type": "article",
    "title": "Exact values F(6)=14 and F(7)=17 for a function of Gowers and Long",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          22
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-2HHHW35HE98QVA5B",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-2HHHW35HE98QVA5B/versions/v1/",
    "note": "Research preprint; ARR-2026-2HHHW35HE98QVA5B v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "For triples a,b in Z^3 write a <_2 b if a_i < b_i in at least two coordinates. Let F(n) be the maximal length of a sequence of triples in {1,...,n}^3 with a^i <_2 a^j whenever i<j. Gowers and Long conjecture F(n) <= n^{3/2} for all n and tabulate F(n) for n<=5. Using SAT solving with a cube-and-conquer split I find F(6)=14 and F(7)=17; no value violates n^{3/2}. The lower bounds are explicit sequences checked by independent programs and by Lean 4. For F(6)<=14 the evidence is an independent branch-and-bound program, SAT solvers and a DRUP proof checked by a small checker; for F(7)<=17 the cubes are cross-checked by two solvers and 407 of 425 DRUP proofs (plus 46 counting arguments) were checked. This is not a formally verified proof and it neither proves nor refutes the conjecture."
  }
]