{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/enumerating-k-sat-functions","title":"Enumerating k-SAT functions","arxiv_id":"2107.09233","date":"2021-07-20","proceeding":null,"authors":["Dingding Dong","Nitya Mani","Yufei Zhao"],"abstract":"How many $k$-SAT functions on $n$ boolean variables are there? What does a typical such function look like? Bollob\\'as, Brightwell, and Leader conjectured that, for each fixed $k \\ge 2$, the number of $k$-SAT functions on $n$ variables is $(1+o(1))2^{\\binom{n}{k} + n}$, or equivalently: a $1-o(1)$ fraction of all $k$-SAT functions are unate, i.e., monotone after negating some variables. They proved a weaker version of the conjecture for $k=2$. The conjecture was confirmed for $k=2$ by Allen and $k=3$ by Ilinca and Kahn. We show that the problem of enumerating $k$-SAT functions is equivalent to a Tur\\'an density problem for partially directed hypergraphs. Our proof uses the hypergraph container method. Furthermore, we confirm the Bollob\\'as--Brightwell--Leader conjecture for $k=4$ by solving the corresponding Tur\\'an density problem. Our solution applies a recent result of F\\\"uredi and Maleki on the minimum triangular edge density in a graph of given edge density. In an appendix (by Nitya Mani and Edward Yu), we further confirm the $k=5$ case of the conjecture via a brute force computer search.","url_abs":"https://arxiv.org/abs/2107.09233v3","url_pdf":"https://arxiv.org/pdf/2107.09233v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"enumerating-k-sat-functions","repo_url":"https://github.com/thingarfield/density-k-pdg","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}