{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:53Z","timestamp":1753893833647,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A classic result of Marcus and Tardos (previously known as the Stanley-Wilf conjecture) bounds from above the number of $n$-permutations ($\\sigma \\in S_n$) that do not contain a specific sub-permutation. In particular, it states that for any fixed permutation $\\pi$, the number of $n$-permutations that avoid $\\pi$ is at most exponential in $n$. In this paper, we generalize this result. We bound the number of avoidant $n$-permutations even if they only have to avoid $\\pi$ at specific indices. We consider a $k$-uniform hypergraph $\\Lambda$ on $n$ vertices and count the $n$-permutations that avoid $\\pi$ at the indices corresponding to the edges of $\\Lambda$. We analyze both the random and deterministic hypergraph cases. This problem was originally proposed by Asaf Ferber.\r\nWhen $\\Lambda$ is a random hypergraph with edge density $\\alpha$, we show that the expected number of $\\Lambda$-avoiding $n$-permutations is bounded (both upper and lower) as $\\exp(O(n))\\alpha^{-\\frac{n}{k-1}}$, using a supersaturation version of F\\\"{u}redi-Hajnal.\r\nIn the deterministic case we show that, for $\\Lambda$ containing many size $L$ cliques, the number of $\\Lambda$-avoiding $n$-permutations is $O\\left(\\frac{n\\log^{2+\\epsilon}n}{L}\\right)^n$, giving a nontrivial bound with $L$ polynomial in $n$. Our main tool in the analysis of this deterministic case is the new and revolutionary hypergraph containers method, developed in papers of Balogh-Morris-Samotij and Saxton-Thomason.<\/jats:p>","DOI":"10.37236\/9014","type":"journal-article","created":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T01:49:04Z","timestamp":1639705744000},"source":"Crossref","is-referenced-by-count":0,"title":["Pattern Avoidance Over a Hypergraph"],"prefix":"10.37236","volume":"28","author":[{"given":"Benjamin","family":"Gunby","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maxwell","family":"Fishelson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2021,12,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p52\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T01:49:04Z","timestamp":1639705744000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i4p52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,17]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,10,8]]}},"URL":"https:\/\/doi.org\/10.37236\/9014","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2021,12,17]]},"article-number":"P4.52"}}