Proof-producing SAT solving of open combinatorial instances

Settle open finite combinatorial questions with SAT solvers that emit checkable unsatisfiability proofs (DRAT/LRAT). Examples of solved cases are Boolean Pythagorean triples, Schur number five, Keller's conjecture in dimension 7, and the empty hexagon number.

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Cite

@misc{cairn-sat-hard-combinatorial-instances,
  title        = {Proof-producing SAT solving of open combinatorial instances},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/sat-hard-combinatorial-instances}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

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On the literature board

The problem

Many finite combinatorial questions reduce to the satisfiability of a propositional formula. Examples are Ramsey-type colourings, Schur and van der Waerden numbers, and point configurations. A satisfying assignment is a short certificate. Unsatisfiability needs a clausal proof (DRAT, LRAT, LPR) that a checker validates, ideally a formally verified one such as cake_lpr.

Known status. Heule, Kullmann and Marek (2016) showed with cube-and-conquer that {1,…,7824} can be 2-coloured without a monochromatic Pythagorean triple but {1,…,7825} cannot. The original proof was about 200 TB. Heule (2017) determined Schur number five. Brakensiek, Heule, Mackey and Narváez (2020) settled Keller's conjecture in dimension 7, certifying the proof with a formally verified checker. Heule and Scheucher (2024) showed that every 30 points in general position contain an empty hexagon, with LRAT proofs checked by cakeLPR. Subercaseaux et al. then verified the encoding in Lean (ITP 2024). The SAT Competition requires proofs for UNSAT claims in its main track. Open targets include the sixth Schur number and other small Ramsey-type values.

What counts as progress

A new value or bound for a stated open instance, with the encoding, solver logs and a proof checked by a verified checker.
Lean or other formal verification that an encoding faithfully represents the mathematical statement, which is often the weakest link.
Better encodings or symmetry breaking that shrink known proofs, with benchmarks.
Documented negative results: an encoding that does not finish within stated resources.

How it is checked. Satisfying assignments are checked directly against the CNF. UNSAT claims are checked by re-running a verified proof checker on the published CNF and proof. Reviewers check the encoding against the mathematical statement, unless it is formally verified.

Current state

The problem