Erdős-Selfridge Theorem for Nonmonotone CNFs

Authors Md Lutfar Rahman, Thomas Watson

Thumbnail PDF


  • Filesize: 0.58 MB
  • 11 pages

Document Identifiers

Author Details

Md Lutfar Rahman
  • Amazon, Seattle, WA, USA
Thomas Watson
  • University of Memphis, TN, USA


We thank anonymous reviewers for their comments.

Cite AsGet BibTex

Md Lutfar Rahman and Thomas Watson. Erdős-Selfridge Theorem for Nonmonotone CNFs. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 31:1-31:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


In an influential paper, Erdős and Selfridge introduced the Maker-Breaker game played on a hypergraph, or equivalently, on a monotone CNF. The players take turns assigning values to variables of their choosing, and Breaker’s goal is to satisfy the CNF, while Maker’s goal is to falsify it. The Erdős-Selfridge Theorem says that the least number of clauses in any monotone CNF with k literals per clause where Maker has a winning strategy is Θ(2^k). We study the analogous question when the CNF is not necessarily monotone. We prove bounds of Θ(√2 ^k) when Maker plays last, and Ω(1.5^k) and O(r^k) when Breaker plays last, where r = (1+√5)/2≈ 1.618 is the golden ratio.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Game
  • nonmonotone
  • CNFs


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Lauri Ahlroth and Pekka Orponen. Unordered constraint satisfaction games. In Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 64-75. Springer, 2012. Google Scholar
  2. Jesper Byskov. Maker-Maker and Maker-Breaker games are PSPACE-complete. Technical Report RS-04-14, BRICS, Department of Computer Science, Aarhus University, 2004. Google Scholar
  3. Paul Erdős and John Selfridge. On a combinatorial game. Journal of Combinatorial Theory, Series A, 14(3), 1973. Google Scholar
  4. Dan Hefetz, Michael Krivelevich, Miloš Stojaković, and Tibor Szabó. Positional Games. Springer, 2014. Google Scholar
  5. Martin Kutz. The Angel Problem, Positional Games, and Digraph Roots. PhD thesis, Freie Universität Berlin, 2004. Chapter 2: Weak Positional Games. Google Scholar
  6. Martin Kutz. Weak positional games on hypergraphs of rank three. In Proceedings of the 3rd European Conference on Combinatorics, Graph Theory, and Applications (EuroComb), pages 31-36. Discrete Mathematics & Theoretical Computer Science, 2005. Google Scholar
  7. Md Lutfar Rahman and Thomas Watson. Complexity of unordered CNF games. ACM Transactions on Computation Theory, 12(3):18:1-18:18, 2020. Google Scholar
  8. Md Lutfar Rahman and Thomas Watson. Tractable unordered 3-CNF games. In Proceedings of the 14th Latin American Theoretical Informatics Symposium (LATIN), pages 360-372. Springer, 2020. Google Scholar
  9. Md Lutfar Rahman and Thomas Watson. 6-uniform Maker-Breaker game is PSPACE-complete. In Proceedings of the 38th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 57:1-57:15. Schloss Dagstuhl, 2021. Google Scholar
  10. Thomas Schaefer. Complexity of decision problems based on finite two-person perfect-information games. In Proceedings of the 8th Symposium on Theory of Computing (STOC), pages 41-49. ACM, 1976. Google Scholar
  11. Thomas Schaefer. On the complexity of some two-person perfect-information games. Journal of Computer and System Sciences, 16(2):185-225, 1978. Google Scholar