{"extern":"1","title":"Hypergraph cuts above the average","user_id":"6785fbc1-c503-11eb-8a32-93094b40e1cf","month":"08","day":"01","status":"public","abstract":[{"text":"An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size m/2+Ω)(m−−√) and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is Ω(m−−√) larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k ≥ 4 or r ≥ 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is Ω(m5/9) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.","lang":"eng"}],"_id":"9580","oa_version":"Preprint","external_id":{"arxiv":["1803.08462"]},"quality_controlled":"1","date_updated":"2023-02-23T14:01:41Z","intvolume":"       233","date_published":"2019-08-01T00:00:00Z","publication":"Israel Journal of Mathematics","article_processing_charge":"No","scopus_import":"1","fulldoi":"https://doi.org/10.1007/s11856-019-1897-z","doi":"10.1007/s11856-019-1897-z","publication_status":"published","date_created":"2021-06-21T13:36:02Z","article_type":"original","year":"2019","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1803.08462"}],"issue":"1","type":"journal_article","author":[{"first_name":"David","full_name":"Conlon, David","last_name":"Conlon"},{"full_name":"Fox, Jacob","first_name":"Jacob","last_name":"Fox"},{"id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","orcid":"0000-0002-4003-7567","full_name":"Kwan, Matthew Alan","first_name":"Matthew Alan","last_name":"Kwan"},{"last_name":"Sudakov","first_name":"Benny","full_name":"Sudakov, Benny"}],"citation":{"apa":"Conlon, D., Fox, J., Kwan, M. A., &#38; Sudakov, B. (2019). Hypergraph cuts above the average. <i>Israel Journal of Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s11856-019-1897-z\">https://doi.org/10.1007/s11856-019-1897-z</a>","mla":"Conlon, David, et al. “Hypergraph Cuts above the Average.” <i>Israel Journal of Mathematics</i>, vol. 233, no. 1, Springer, 2019, pp. 67–111, doi:<a href=\"https://doi.org/10.1007/s11856-019-1897-z\">10.1007/s11856-019-1897-z</a>.","ista":"Conlon D, Fox J, Kwan MA, Sudakov B. 2019. Hypergraph cuts above the average. Israel Journal of Mathematics. 233(1), 67–111.","chicago":"Conlon, David, Jacob Fox, Matthew Alan Kwan, and Benny Sudakov. “Hypergraph Cuts above the Average.” <i>Israel Journal of Mathematics</i>. Springer, 2019. <a href=\"https://doi.org/10.1007/s11856-019-1897-z\">https://doi.org/10.1007/s11856-019-1897-z</a>.","ama":"Conlon D, Fox J, Kwan MA, Sudakov B. Hypergraph cuts above the average. <i>Israel Journal of Mathematics</i>. 2019;233(1):67-111. doi:<a href=\"https://doi.org/10.1007/s11856-019-1897-z\">10.1007/s11856-019-1897-z</a>","ieee":"D. Conlon, J. Fox, M. A. Kwan, and B. Sudakov, “Hypergraph cuts above the average,” <i>Israel Journal of Mathematics</i>, vol. 233, no. 1. Springer, pp. 67–111, 2019.","short":"D. Conlon, J. Fox, M.A. Kwan, B. Sudakov, Israel Journal of Mathematics 233 (2019) 67–111."},"publisher":"Springer","page":"67-111","arxiv":1,"oa":1,"language":[{"iso":"eng"}],"volume":233,"publication_identifier":{"eissn":["1565-8511"],"issn":["0021-2172"]}}