Combinatorics, Probability and Computing最新文献

{"title":"Polarised random -SAT","authors":"Joel Larsson Danielsson, Klas Markström","doi":"10.1017/s0963548323000226","DOIUrl":"https://doi.org/10.1017/s0963548323000226","url":null,"abstract":"<p>In this paper we study a variation of the random <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000226:S0963548323000226_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-SAT problem, called polarised random <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000226:S0963548323000226_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-SAT, which contains both the classical random <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000226:S0963548323000226_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-SAT model and the random version of monotone <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000226:S0963548323000226_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-SAT another well-known NP-complete version of SAT. In this model there is a polarisation parameter <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000226:S0963548323000226_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>, and in half of the clauses each variable occurs negated with probability <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000226:S0963548323000226_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span> and pure otherwise, while in the other half the probabilities are interchanged. For <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000226:S0963548323000226_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$p=1/2$</span></span></img></span></span> we get the classical random <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000226:S0963548323000226_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-SAT model, and at the other extreme we have the fully polarised model where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000226:S0963548323000226","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The codegree Turán density of tight cycles minus one edge","authors":"Simón Piga, Marcelo Sales, Bjarne Schülke","doi":"10.1017/s0963548323000196","DOIUrl":"https://doi.org/10.1017/s0963548323000196","url":null,"abstract":"<p>Given <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000196:S0963548323000196_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$alpha gt 0$</span></span></img></span></span> and an integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000196:S0963548323000196_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$ell geq 5$</span></span></img></span></span>, we prove that every sufficiently large <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000196:S0963548323000196_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$3$</span></span></img></span></span>-uniform hypergraph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000196:S0963548323000196_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$H$</span></span></img></span></span> on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000196:S0963548323000196_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span> vertices in which every two vertices are contained in at least <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000196:S0963548323000196_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$alpha n$</span></span></img></span></span> edges contains a copy of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000196:S0963548323000196_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$C_ell ^{-}$</span></span></img></span></span>, a tight cycle on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000196:S0963548323000196_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$ell$</span></span></img></span></span> vertices minus one edge. This improves a previous result by Balogh, Clemen, and Lidický.</p>","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Unavoidable patterns in locally balanced colourings","authors":"Nina Kamčev, Alp Müyesser","doi":"10.1017/s0963548323000160","DOIUrl":"https://doi.org/10.1017/s0963548323000160","url":null,"abstract":"\u0000\t <jats:p>Which patterns must a two-colouring of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000160_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$K_n$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> contain if each vertex has at least <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000160_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$varepsilon n$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> red and <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000160_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$varepsilon n$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> blue neighbours? We show that when <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000160_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$varepsilon gt 1/4$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000160_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$K_n$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> must contain a complete subgraph on <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000160_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$Omega (log n)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> vertices where one of the colours forms a balanced complete bipartite graph.</jats:p>\u0000\t <jats:p>When <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000160_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$varepsilon leq 1/4$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, this statement is no longer true, as evidenced by the following colouring <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000160_inline8.png\" />\u0000\t\t<jats:tex-math>\u0000$chi$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000160_inline9.png\" />\u0000\t\t<jats:tex-math>\u0000$K_n$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. Divide the vertex set into <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xml","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85325546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The bunkbed conjecture holds in the limit","authors":"Tom Hutchcroft, Alexander Kent, Petar Nizić-Nikolac","doi":"10.1017/s096354832200027x","DOIUrl":"https://doi.org/10.1017/s096354832200027x","url":null,"abstract":"<p>Let <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S096354832200027X:S096354832200027X_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$G=(V,E)$\u0000</span></span>\u0000</span>\u0000</span> be a countable graph. The Bunkbed graph of <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S096354832200027X:S096354832200027X_inline3.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$G$\u0000</span></span>\u0000</span>\u0000</span> is the product graph <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S096354832200027X:S096354832200027X_inline4.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$G times K_2$\u0000</span></span>\u0000</span>\u0000</span>, which has vertex set <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S096354832200027X:S096354832200027X_inline5.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$Vtimes {0,1}$\u0000</span></span>\u0000</span>\u0000</span> with “horizontal” edges inherited from <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S096354832200027X:S096354832200027X_inline6.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$G$\u0000</span></span>\u0000</span>\u0000</span> and additional “vertical” edges connecting <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S096354832200027X:S096354832200027X_inline7.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$(w,0)$\u0000</span></span>\u0000</span>\u0000</span> and <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S096354832200027X:S096354832200027X_inline8.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$(w,1)$\u0000</span></span>\u0000</span>\u0000</span> for each <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S096354832200027X:S096354832200027X_inline9.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$w in V$\u0000</span></span>\u0000</span>\u0000</span>. Kasteleyn’s Bunkbed conjecture states that for each <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S096354832200027X:S096354832200027X_inline10.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$u,v in V$\u0000</span></span>\u0000</span>\u0000</span> and <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:2023040","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A bipartite version of the Erdős–McKay conjecture","authors":"Eoin Long, Laurenţiu Ploscaru","doi":"10.1017/s0963548322000347","DOIUrl":"https://doi.org/10.1017/s0963548322000347","url":null,"abstract":"<p>An old conjecture of Erdős and McKay states that if all homogeneous sets in an <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000347:S0963548322000347_inline1.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$n$\u0000</span></span>\u0000</span>\u0000</span>-vertex graph are of order <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000347:S0963548322000347_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$O(!log n)$\u0000</span></span>\u0000</span>\u0000</span> then the graph contains induced subgraphs of each size from <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000347:S0963548322000347_inline3.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000${0,1,ldots, Omega big(n^2big)}$\u0000</span></span>\u0000</span>\u0000</span>. We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000347:S0963548322000347_inline4.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$n times n$\u0000</span></span>\u0000</span>\u0000</span> bipartite graph are of order <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000347:S0963548322000347_inline5.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$O(!log n)$\u0000</span></span>\u0000</span>\u0000</span>, then the graph contains induced subgraphs of each size from <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000347:S0963548322000347_inline6.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000${0,1,ldots, Omega big(n^2big)}$\u0000</span></span>\u0000</span>\u0000</span>.</p>","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fluctuations of subgraph counts in graphon based random graphs","authors":"Bhaswar B. Bhattacharya, Anirban Chatterjee, Svante Janson","doi":"10.1017/s0963548322000335","DOIUrl":"https://doi.org/10.1017/s0963548322000335","url":null,"abstract":"<p>Given a graphon <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000335:S0963548322000335_inline1.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$W$\u0000</span></span>\u0000</span>\u0000</span> and a finite simple graph <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000335:S0963548322000335_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$H$\u0000</span></span>\u0000</span>\u0000</span>, with vertex set <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000335:S0963548322000335_inline3.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$V(H)$\u0000</span></span>\u0000</span>\u0000</span>, denote by <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000335:S0963548322000335_inline4.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$X_n(H, W)$\u0000</span></span>\u0000</span>\u0000</span> the number of copies of <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000335:S0963548322000335_inline5.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$H$\u0000</span></span>\u0000</span>\u0000</span> in a <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000335:S0963548322000335_inline6.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$W$\u0000</span></span>\u0000</span>\u0000</span>-random graph on <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000335:S0963548322000335_inline7.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$n$\u0000</span></span>\u0000</span>\u0000</span> vertices. The asymptotic distribution of <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000335:S0963548322000335_inline8.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$X_n(H, W)$\u0000</span></span>\u0000</span>\u0000</span> was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000335:S0963548322000335_inline9.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$H$\u0000</span></span>\u0000</span>\u0000</span> is a clique. In this paper, we extend this result to any fixed graph <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Supercritical site percolation on the hypercube: small components are small","authors":"Sahar Diskin, Michael Krivelevich","doi":"10.1017/s0963548322000323","DOIUrl":"https://doi.org/10.1017/s0963548322000323","url":null,"abstract":"<p>We consider supercritical site percolation on the <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000323:S0963548322000323_inline1.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$d$\u0000</span></span>\u0000</span>\u0000</span>-dimensional hypercube <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000323:S0963548322000323_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$Q^d$\u0000</span></span>\u0000</span>\u0000</span>. We show that typically all components in the percolated hypercube, besides the giant, are of size <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230406132554462-0824:S0963548322000323:S0963548322000323_inline3.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$O(d)$\u0000</span></span>\u0000</span>\u0000</span>. This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.</p>","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"6 2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New upper bounds for the Erdős-Gyárfás problem on generalized Ramsey numbers","authors":"Alex Cameron, Emily Heath","doi":"10.1017/s0963548322000293","DOIUrl":"https://doi.org/10.1017/s0963548322000293","url":null,"abstract":"A <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline1.png\" /><jats:tex-math> $(p,q)$ </jats:tex-math></jats:alternatives></jats:inline-formula>-colouring of a graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline2.png\" /><jats:tex-math> $G$ </jats:tex-math></jats:alternatives></jats:inline-formula> is an edge-colouring of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline3.png\" /><jats:tex-math> $G$ </jats:tex-math></jats:alternatives></jats:inline-formula> which assigns at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline4.png\" /><jats:tex-math> $q$ </jats:tex-math></jats:alternatives></jats:inline-formula> colours to each <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline5.png\" /><jats:tex-math> $p$ </jats:tex-math></jats:alternatives></jats:inline-formula>-clique. The problem of determining the minimum number of colours, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline6.png\" /><jats:tex-math> $f(n,p,q)$ </jats:tex-math></jats:alternatives></jats:inline-formula>, needed to give a <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline7.png\" /><jats:tex-math> $(p,q)$ </jats:tex-math></jats:alternatives></jats:inline-formula>-colouring of the complete graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline8.png\" /><jats:tex-math> $K_n$ </jats:tex-math></jats:alternatives></jats:inline-formula> is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline9.png\" /><jats:tex-math> $r_k(p)$ </jats:tex-math></jats:alternatives></jats:inline-formula>. The best-known general upper bound on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000293_inline10.png\" /><jats:tex-math> $f(n,p,q)$ </jats:tex-math></jats:alternatives></jats:inline-formula> was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where <jats:inline-formula><jats:alternatives><jat","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"124 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On mappings on the hypercube with small average stretch","authors":"Lucas Boczkowski, Igor Shinkar","doi":"10.1017/s0963548322000281","DOIUrl":"https://doi.org/10.1017/s0963548322000281","url":null,"abstract":"Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000281_inline1.png\" /><jats:tex-math> $A subseteq {0,1}^n$ </jats:tex-math></jats:alternatives></jats:inline-formula> be a set of size <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000281_inline2.png\" /><jats:tex-math> $2^{n-1}$ </jats:tex-math></jats:alternatives></jats:inline-formula>, and let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000281_inline3.png\" /><jats:tex-math> $phi ,:, {0,1}^{n-1} to A$ </jats:tex-math></jats:alternatives></jats:inline-formula> be a bijection. We define <jats:italic>the average stretch</jats:italic> of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000281_inline4.png\" /><jats:tex-math> $phi$ </jats:tex-math></jats:alternatives></jats:inline-formula> as<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548322000281_eqnU1.png\" /><jats:tex-math> begin{equation*} {sf avgStretch}(phi ) = {mathbb E}[{{sf dist}}(phi (x),phi (x'))], end{equation*} </jats:tex-math></jats:alternatives></jats:disp-formula>where the expectation is taken over uniformly random <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000281_inline5.png\" /><jats:tex-math> $x,x' in {0,1}^{n-1}$ </jats:tex-math></jats:alternatives></jats:inline-formula> that differ in exactly one coordinate.In this paper, we continue the line of research studying mappings on the discrete hypercube with small average stretch. We prove the following results.<jats:list list-type=\"bullet\"><jats:list-item>For any set <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000281_inline6.png\" /><jats:tex-math> $A subseteq {0,1}^n$ </jats:tex-math></jats:alternatives></jats:inline-formula> of density <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000281_inline7.png\" /><jats:tex-math> $1/2$ </jats:tex-math></jats:alternatives></jats:inline-formula> there exists a bijection <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000281_inline8.png\" /><jats:tex-math> $phi _A ,:, {0,1}^{n-1} to A$ </jats:tex-math></jats:alternatives></jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Towards the 0-statement of the Kohayakawa-Kreuter conjecture","authors":"Joseph Hyde","doi":"10.1017/s0963548322000219","DOIUrl":"https://doi.org/10.1017/s0963548322000219","url":null,"abstract":"<p>In this paper, we study asymmetric Ramsey properties of the random graph <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230206124632469-0262:S0963548322000219:S0963548322000219_inline1.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$G_{n,p}$\u0000</span></span>\u0000</span>\u0000</span>. Let <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230206124632469-0262:S0963548322000219:S0963548322000219_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$r in mathbb{N}$\u0000</span></span>\u0000</span>\u0000</span> and <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230206124632469-0262:S0963548322000219:S0963548322000219_inline3.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$H_1, ldots, H_r$\u0000</span></span>\u0000</span>\u0000</span> be graphs. We write <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230206124632469-0262:S0963548322000219:S0963548322000219_inline4.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$G_{n,p} to (H_1, ldots, H_r)$\u0000</span></span>\u0000</span>\u0000</span> to denote the property that whenever we colour the edges of <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230206124632469-0262:S0963548322000219:S0963548322000219_inline5.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$G_{n,p}$\u0000</span></span>\u0000</span>\u0000</span> with colours from the set <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230206124632469-0262:S0963548322000219:S0963548322000219_inline6.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$[r] ,{:!=}, {1, ldots, r}$\u0000</span></span>\u0000</span>\u0000</span> there exists <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230206124632469-0262:S0963548322000219:S0963548322000219_inline7.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$i in [r]$\u0000</span></span>\u0000</span>\u0000</span> and a copy of <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230206124632469-0262:S0963548322000219:S0963548322000219_inline8.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$H_i$\u0000</span></span>\u0000</span>\u0000</span> in <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230206124632469-0262:S0963548322000219:S0963548322000219_inline9.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$G_{n,p}$\u0000</span></span>\u0000</span>\u0000</span> monochromatic in colour <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}