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Discrete Math."],"published-print":{"date-parts":[[2023,9,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Recently Chase determined the maximum possible number of cliques of size [Formula: see text] in a graph on [Formula: see text] vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have [Formula: see text] edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For [Formula: see text]-graphs with [Formula: see text] a number of issues arise that do not appear in the graph case. For instance, for general [Formula: see text]-graphs we can assign degrees to any [Formula: see text]-subset of the vertex set with [Formula: see text]. We establish bounds on the number of [Formula: see text]-cliques in an [Formula: see text]-graph [Formula: see text] with [Formula: see text]-degree bounded by [Formula: see text] in three contexts: [Formula: see text] has [Formula: see text] vertices; [Formula: see text] has [Formula: see text] (hyper)edges; and (generalizing the previous case) [Formula: see text] has a fixed number [Formula: see text] of [Formula: see text]-cliques for some [Formula: see text] with [Formula: see text]. When [Formula: see text] is of a special form we characterize the extremal [Formula: see text]-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or partial Steiner systems. On the way to proving our uniqueness results, we extend results of F\u00fcredi and Griggs on uniqueness in Kruskal\u2013Katona from the shadow case to the clique case.<\/jats:p>","DOI":"10.1137\/22m1507565","type":"journal-article","created":{"date-parts":[[2023,7,13]],"date-time":"2023-07-13T17:20:38Z","timestamp":1689268838000},"page":"1436-1456","source":"Crossref","is-referenced-by-count":0,"title":["Many Cliques in Bounded-Degree Hypergraphs"],"prefix":"10.1137","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6633-3444","authenticated-orcid":true,"given":"Rachel","family":"Kirsch","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jamie","family":"Radcliffe","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,7,12]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2016.03.004"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2021.05.005"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.37236\/10972"},{"key":"ref4","author":"Chase Z.","year":"2020","journal-title":"Adv. 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