{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:34:19Z","timestamp":1787322859712,"version":"build-2736575974"},"reference-count":16,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2023,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Lionel Levine\u2019s hat challenge has [Formula: see text] players, each with a (very large or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their respective stacks. They are proclaimed successful only if they are all correct. Levine\u2019s conjecture is that the success probability tends to zero when the number of players grows. We prove that this success probability is strictly decreasing in the number of players, and present some connections to problems in graph theory: relating the size of the largest independent set in a graph and in a random induced subgraph of it, and bounding the size of a set of vertices intersecting every maximum-size independent set in a graph.<\/jats:p>","DOI":"10.1137\/22m1515975","type":"journal-article","created":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T06:02:45Z","timestamp":1700460165000},"page":"2717-2729","source":"Crossref","is-referenced-by-count":2,"title":["The Success Probability in Levine\u2019s Hat Problem, and Independent Sets in Graphs"],"prefix":"10.1137","volume":"37","author":[{"given":"Noga","family":"Alon","sequence":"first","affiliation":[{"name":"Princeton University, Princeton, NJ 08544 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ehud","family":"Friedgut","sequence":"additional","affiliation":[{"name":"Math, Weizmann Institute of Science, Rehovot 7610001, Israel."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gil","family":"Kalai","sequence":"additional","affiliation":[{"name":"Hebrew University, Jerusalem 91905, Israel."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3384-3720","authenticated-orcid":true,"given":"Guy","family":"Kindler","sequence":"additional","affiliation":[{"name":"Hebrew University, Jerusalem 91905, Israel."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,11,20]]},"reference":[{"key":"ref1","volume-title":"Graph Coloring: Local and Global, Public Lecture","author":"Alon N.","year":"2017"},{"key":"ref2","doi-asserted-by":"crossref","unstructured":"N. Alon, A. Hassidim, E. Lubetzky, U. Stav, and A. Weinstein, Broadcasting with side information, in Proceeding of the 49th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 823\u2013832.","DOI":"10.1109\/FOCS.2008.41"},{"key":"ref3","volume-title":"The Probabilistic Method","author":"Alon N.","year":"2016","edition":"4"},{"key":"ref4","unstructured":"N. Alon and G. Tardos, private communication."},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2014-00816-X"},{"key":"ref6","doi-asserted-by":"crossref","unstructured":"J. Buhler, C. Freiling, R. Graham, J. Kariv, J. R. Roche, M. Tiefenbruck, C. Van Alten, and D. Yeroshkin, On Levine\u2019s Notorious Hat Puzzle, preprint, https:\/\/arxiv.org\/abs\/1407.4711, 2021.","DOI":"10.1515\/9783110754216-008"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1201\/9781439863879"},{"key":"ref8","unstructured":"Z. Dong and Z. Wu, On the Stability of Graph Independence Number, preprint, https:\/\/arxiv.org\/abs\/2102.13306, 2021."},{"key":"ref9","series-title":"Bolyai Soc. Math. Stud. 3","first-page":"217","volume-title":"Extremal Problems for Finite Sets (Visegr\u00e1d, 1991)","author":"Erd\u0151s P.","year":"1994"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20412"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1965-072-1"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100034241"},{"key":"ref13","unstructured":"T. Khovanova, How many hats can fit on your head? 2011, https:\/\/blog.tanyakhovanova.com\/2011\/04."},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9800(66)80027-3"},{"key":"ref15","unstructured":"L. Rabern, On Hitting All Maximum Cliques with an Independent Set, preprint, https:\/\/arxiv.org\/abs\/0907.3705, 2009."},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-014-0562-8"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1515975","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:29:51Z","timestamp":1787318991000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1515975"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,20]]},"references-count":16,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,12,31]]}},"alternative-id":["10.1137\/22M1515975"],"URL":"https:\/\/doi.org\/10.1137\/22m1515975","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,20]]}}}