A SAT-based Resolution of Lam's Problem
This work provides a verifiable resolution to a long-standing problem in projective geometry, addressing the lack of certifiable nonexistence proofs from prior computational searches.
This paper resolves Lam's problem, which concerns the existence of a projective plane of order ten, by translating it into Boolean logic and using SAT solvers. This approach confirmed the nonexistence of such a plane and produced verifiable nonexistence certificates, in contrast to previous specialized searches.
In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometry$\unicode{x2014}$the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam's problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searches$\unicode{x2014}$highlighting the difficulty of relying on special-purpose search code for nonexistence results.