{"title":"Intersections of iterated shadows","authors":"Hou Tin Chau, David Ellis, Marius Tiba","doi":"arxiv-2409.05487","DOIUrl":null,"url":null,"abstract":"We show that if $\\mathcal{A} \\subset {[n] \\choose n/2}$ with measure bounded\naway from zero and from one, then the $\\Omega(\\sqrt{n})$-iterated upper shadows\nof $\\mathcal{A}$ and $\\mathcal{A}^c$ intersect in a set of positive measure.\nThis confirms (in a strong form) a conjecture of Friedgut. It can be seen as a\nstability result for the Kruskal--Katona theorem.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05487","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We show that if $\mathcal{A} \subset {[n] \choose n/2}$ with measure bounded
away from zero and from one, then the $\Omega(\sqrt{n})$-iterated upper shadows
of $\mathcal{A}$ and $\mathcal{A}^c$ intersect in a set of positive measure.
This confirms (in a strong form) a conjecture of Friedgut. It can be seen as a
stability result for the Kruskal--Katona theorem.
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