{"department":[{"_id":"MaKw"}],"type":"journal_article","day":"11","month":"09","publisher":"Oxford University Press","has_accepted_license":"1","PlanS_conform":"1","OA_place":"publisher","abstract":[{"text":"Let r, k,  be integers such that 0 ≤  ≤ (k/r). Given a large r-uniform hypergraph G, we consider the\r\nfraction of k-vertex subsets that span exactly  edges. If  is 0 or (k/r), this fraction can be exactly 1 (by taking G to be empty or complete), but for all other values of , one might suspect that this fraction is always significantly smaller than 1.\r\nIn this paper we prove an essentially optimal result along these lines: if  is not 0 or (k/r), then this\r\nfraction is at most (1/e) + ε, assuming k is sufficiently large in terms of r and ε > 0, and G is sufficiently large in terms of k. Previously, this was only known for a very limited range of values of r, k,  (due to Kwan–Sudakov–Tran, Fox–Sauermann, and Martinsson–Mousset–Noever–Trujic). Our result answers a question of Alon–Hefetz–Krivelevich–Tyomkyn, who suggested this as a hypergraph generalization of their edge-statistics conjecture. We also prove a much stronger bound when  is far from 0 and (k/r).","lang":"eng"}],"year":"2025","author":[{"last_name":"Jain","first_name":"Vishesh","full_name":"Jain, Vishesh"},{"first_name":"Matthew Alan","orcid":"0000-0002-4003-7567","full_name":"Kwan, Matthew Alan","last_name":"Kwan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3"},{"last_name":"Mubayi","full_name":"Mubayi, Dhruv","first_name":"Dhruv"},{"full_name":"Tran, Tuan","first_name":"Tuan","last_name":"Tran"}],"article_number":"rnaf273","date_created":"2025-10-20T11:08:57Z","publication":"International Mathematics Research Notices","status":"public","OA_type":"hybrid","publication_status":"published","intvolume":"      2025","_id":"20504","corr_author":"1","volume":2025,"issue":"18","article_processing_charge":"Yes (via OA deal)","language":[{"iso":"eng"}],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"quality_controlled":"1","license":"https://creativecommons.org/licenses/by/4.0/","oa_version":"Published Version","file_date_updated":"2025-10-21T07:36:56Z","external_id":{"arxiv":["2505.03954"]},"arxiv":1,"project":[{"grant_number":"101076777","_id":"bd95085b-d553-11ed-ba76-e55d3349be45","name":"Randomness and structure in combinatorics"}],"scopus_import":"1","citation":{"mla":"Jain, Vishesh, et al. “The Edge-Statistics Conjecture for Hypergraphs.” <i>International Mathematics Research Notices</i>, vol. 2025, no. 18, rnaf273, Oxford University Press, 2025, doi:<a href=\"https://doi.org/10.1093/imrn/rnaf273\">10.1093/imrn/rnaf273</a>.","short":"V. Jain, M.A. Kwan, D. Mubayi, T. Tran, International Mathematics Research Notices 2025 (2025).","ama":"Jain V, Kwan MA, Mubayi D, Tran T. The edge-statistics conjecture for hypergraphs. <i>International Mathematics Research Notices</i>. 2025;2025(18). doi:<a href=\"https://doi.org/10.1093/imrn/rnaf273\">10.1093/imrn/rnaf273</a>","ista":"Jain V, Kwan MA, Mubayi D, Tran T. 2025. The edge-statistics conjecture for hypergraphs. International Mathematics Research Notices. 2025(18), rnaf273.","chicago":"Jain, Vishesh, Matthew Alan Kwan, Dhruv Mubayi, and Tuan Tran. “The Edge-Statistics Conjecture for Hypergraphs.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2025. <a href=\"https://doi.org/10.1093/imrn/rnaf273\">https://doi.org/10.1093/imrn/rnaf273</a>.","apa":"Jain, V., Kwan, M. A., Mubayi, D., &#38; Tran, T. (2025). The edge-statistics conjecture for hypergraphs. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rnaf273\">https://doi.org/10.1093/imrn/rnaf273</a>","ieee":"V. Jain, M. A. Kwan, D. Mubayi, and T. Tran, “The edge-statistics conjecture for hypergraphs,” <i>International Mathematics Research Notices</i>, vol. 2025, no. 18. Oxford University Press, 2025."},"date_updated":"2025-10-21T07:39:40Z","date_published":"2025-09-11T00:00:00Z","file":[{"date_updated":"2025-10-21T07:36:56Z","file_size":774323,"file_name":"2025_IMRN_Jain.pdf","date_created":"2025-10-21T07:36:56Z","file_id":"20511","checksum":"016aa4df9453dc180ae7504ac77bf72f","success":1,"content_type":"application/pdf","creator":"dernst","access_level":"open_access","relation":"main_file"}],"ddc":["510"],"oa":1,"article_type":"original","acknowledgement":"This work was supported by NSF CAREER award DMS-2237646 [to V.J.], ERC Starting Grant “RANDSTRUCT” [no. 101076777 to M.K.], NSF grant DMS-2153576 [to D.M.], and the National Key Research and Development Program of China [2023YFA101020 to T.T.].\r\nWe would like to thank Lisa Sauermann for her helpful comments. We would also like to thank Alex Grebennikov for identifying an oversight in the application of Theorem 7.1 (in a previous version of this paper).","title":"The edge-statistics conjecture for hypergraphs","publication_identifier":{"eissn":["1687-0247"],"issn":["1073-7928"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.1093/imrn/rnaf273"}