Combinatorics, Probability and Computing最新文献

{"title":"On a conjecture of Conlon, Fox, and Wigderson","authors":"Chunchao Fan, Qizhong Lin, Yuanhui Yan","doi":"10.1017/s0963548324000026","DOIUrl":"https://doi.org/10.1017/s0963548324000026","url":null,"abstract":"For graphs <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000026_inline1.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000026_inline2.png\" /> <jats:tex-math> $H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the Ramsey number <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000026_inline3.png\" /> <jats:tex-math> $r(G,H)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the smallest positive integer <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000026_inline4.png\" /> <jats:tex-math> $N$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that any red/blue edge 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=\"S0963548324000026_inline5.png\" /> <jats:tex-math> $K_N$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> contains either a red <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000026_inline6.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> or a blue <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000026_inline7.png\" /> <jats:tex-math> $H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. A book <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000026_inline8.png\" /> <jats:tex-math> $B_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a graph consisting of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000026_inline9.png\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> triangles all sharing a common edge. Recently, Conlon, Fox, and Wigderson conjectured that for any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000026_inline10.png\" /> <jats:tex-math> $0lt alpha lt 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the random lower bound <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139753811","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":"Percolation on irregular high-dimensional product graphs","authors":"Sahar Diskin, Joshua Erde, Mihyun Kang, Michael Krivelevich","doi":"10.1017/s0963548323000469","DOIUrl":"https://doi.org/10.1017/s0963548323000469","url":null,"abstract":"<p>We consider bond percolation on high-dimensional product graphs <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219111306460-0188:S0963548323000469:S0963548323000469_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$G=square _{i=1}^tG^{(i)}$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219111306460-0188:S0963548323000469:S0963548323000469_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$square$</span></span></img></span></span> denotes the Cartesian product. We call the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219111306460-0188:S0963548323000469:S0963548323000469_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$G^{(i)}$</span></span></img></span></span> the base graphs and the product graph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219111306460-0188:S0963548323000469:S0963548323000469_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> the host graph. Very recently, Lichev (<span>J. Graph Theory</span>, 99(4):651–670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219111306460-0188:S0963548323000469:S0963548323000469_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$G_p$</span></span></img></span></span> undergoes a phase transition when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219111306460-0188:S0963548323000469:S0963548323000469_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span> is around <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219111306460-0188:S0963548323000469:S0963548323000469_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$frac{1}{d}$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231219111306460-0188:S0963548323000469:S0963548323000469_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$d$</span></span></img></span></span> is the average degree of the host graph.</p><p>In the supercritical regime, we strengthen Lichev’s result by showing that the giant component is in fact unique, with all other components of order <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"htt","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138820859","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":"Product structure of graph classes with bounded treewidth","authors":"Rutger Campbell, Katie Clinch, Marc Distel, J. Pascal Gollin, Kevin Hendrey, Robert Hickingbotham, Tony Huynh, Freddie Illingworth, Youri Tamitegama, Jane Tan, David R. Wood","doi":"10.1017/s0963548323000457","DOIUrl":"https://doi.org/10.1017/s0963548323000457","url":null,"abstract":"<p>We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the <span>underlying treewidth</span> of a graph class <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207050429486-0539:S0963548323000457:S0963548323000457_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal{G}$</span></span></img></span></span> to be the minimum non-negative integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207050429486-0539:S0963548323000457:S0963548323000457_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$c$</span></span></img></span></span> such that, for some function <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207050429486-0539:S0963548323000457:S0963548323000457_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$f$</span></span></img></span></span>, for every graph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207050429486-0539:S0963548323000457:S0963548323000457_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G in mathcal{G}$</span></span></img></span></span> there is a graph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207050429486-0539:S0963548323000457:S0963548323000457_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$H$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207050429486-0539:S0963548323000457:S0963548323000457_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$textrm{tw}(H) leqslant c$</span></span></img></span></span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207050429486-0539:S0963548323000457:S0963548323000457_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> is isomorphic to a subgraph of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207050429486-0539:S0963548323000457:S0963548323000457_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$H boxtimes K_{f(textrm{tw}(G))}$</span></span></img></span></span>. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547473","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 minimum spanning trees for random Euclidean bipartite graphs","authors":"Mario Correddu, Dario Trevisan","doi":"10.1017/s0963548323000445","DOIUrl":"https://doi.org/10.1017/s0963548323000445","url":null,"abstract":"We consider the minimum spanning tree problem on a weighted complete bipartite graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline1.png\" /> <jats:tex-math> $K_{n_R, n_B}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> whose <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline2.png\" /> <jats:tex-math> $n=n_R+n_B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> vertices are random, i.i.d. uniformly distributed points in the unit cube in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline3.png\" /> <jats:tex-math> $d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> dimensions and edge weights are the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline4.png\" /> <jats:tex-math> $p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-th power of their Euclidean distance, with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline5.png\" /> <jats:tex-math> $pgt 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In the large <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline6.png\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> limit with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline7.png\" /> <jats:tex-math> $n_R/n to alpha _R$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline8.png\" /> <jats:tex-math> $0lt alpha _Rlt 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline9.png\" /> <jats:tex-math> $d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> only. Despite this difference, for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000445_inline10.png\" /> <jats:tex-math> $plt d$ </","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529136","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":"Threshold graphs maximise homomorphism densities","authors":"Grigoriy Blekherman, Shyamal Patel","doi":"10.1017/s096354832300041x","DOIUrl":"https://doi.org/10.1017/s096354832300041x","url":null,"abstract":"Given a fixed graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline1.png\" /> <jats:tex-math> $H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and a constant <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline2.png\" /> <jats:tex-math> $c in [0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we can ask what graphs <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline3.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with edge density <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline4.png\" /> <jats:tex-math> $c$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> asymptotically maximise the homomorphism density of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline5.png\" /> <jats:tex-math> $H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline6.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. For all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline7.png\" /> <jats:tex-math> $H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for which this problem has been solved, the maximum is always asymptotically attained on one of two kinds of graphs: the quasi-star or the quasi-clique. We show that for any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline8.png\" /> <jats:tex-math> $H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> the maximising <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300041X_inline9.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is asymptotically a threshold graph, while the quasi-clique and the quasi-star are the simplest threshold graphs, having only two parts. This result gives us a unified framework to derive a number of results on graph homomorphism maximisation, some of which were also found quite recently and independently using several different approaches. We show that there exist graphs <jats:inline-formu","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529134","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":"Spread-out limit of the critical points for lattice trees and lattice animals in dimensions","authors":"Noe Kawamoto, Akira Sakai","doi":"10.1017/s096354832300038x","DOIUrl":"https://doi.org/10.1017/s096354832300038x","url":null,"abstract":"A spread-out lattice animal is a finite connected set of edges in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline2.png\" /> <jats:tex-math>${{x,y}subset mathbb{Z}^d;:;0lt |x-y|le L}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. A lattice tree is a lattice animal with no loops. The best estimate on the critical point <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline3.png\" /> <jats:tex-math>$p_{textrm{c}}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> so far was achieved by Penrose (<jats:italic>J. Stat. Phys.</jats:italic> 77, 3–15, 1994) : <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline4.png\" /> <jats:tex-math>$p_{textrm{c}}=1/e+O(L^{-2d/7}log L)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for both models for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline5.png\" /> <jats:tex-math>$dge 1$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline6.png\" /> <jats:tex-math>$p_{textrm{c}}=1/e+CL^{-d}+O(L^{-d-1})$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline7.png\" /> <jats:tex-math>$dgt 8$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where the model-dependent constant <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline8.png\" /> <jats:tex-math>$C$</jats:tex-math> </jats:alternatives> </jats:inline-formula> has the random-walk representation <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S096354832300038X_eqnU1.png\" /> <jats:tex-math>begin{align*} C_{textrm{LT}}=sum _{n=2}^infty frac{n+1}{2e}U^{*n}(o),&amp;&amp; C_{textrm{LA}}=C_{textrm{LT}}-frac 1{2e^2}sum _{n=3}^infty U^{*n}(o), end{align*}</jats:tex-math> </jats:alternatives> </jats:disp-formula>where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832300038X_inline9.png\" /> <jats:tex-math>$U^{*n}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the <jats:inline-formula> <jats:alternatives> <","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529135","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":"Large cliques or cocliques in hypergraphs with forbidden order-size pairs","authors":"Maria Axenovich, Domagoj Bradač, Lior Gishboliner, Dhruv Mubayi, Lea Weber","doi":"10.1017/s0963548323000433","DOIUrl":"https://doi.org/10.1017/s0963548323000433","url":null,"abstract":"The well-known Erdős-Hajnal conjecture states that for any graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline1.png\" /> <jats:tex-math> $F$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, there exists <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline2.png\" /> <jats:tex-math> $epsilon gt 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline3.png\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-vertex graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline4.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> that contains no induced copy of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline5.png\" /> <jats:tex-math> $F$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has a homogeneous set of size at least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline6.png\" /> <jats:tex-math> $n^{epsilon }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline7.png\" /> <jats:tex-math> $m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> vertices and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline8.png\" /> <jats:tex-math> $f$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> edges for any positive <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline9.png\" /> <jats:tex-math> $m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000433_inline10.png\" /> <jats:tex-math> $0leq f leq binom{m}{2}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then we o","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529092","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 Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution","authors":"Yassine El Maazouz, Jim Pitman","doi":"10.1017/s0963548323000421","DOIUrl":"https://doi.org/10.1017/s0963548323000421","url":null,"abstract":"The factorially normalized Bernoulli polynomials <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline1.png\" /> <jats:tex-math> $b_n(x) = B_n(x)/n!$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are known to be characterized by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline2.png\" /> <jats:tex-math> $b_0(x) = 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline3.png\" /> <jats:tex-math> $b_n(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline4.png\" /> <jats:tex-math> $n gt 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the anti-derivative of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline5.png\" /> <jats:tex-math> $b_{n-1}(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> subject to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline6.png\" /> <jats:tex-math> $int _0^1 b_n(x) dx = 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We offer a related characterization: <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline7.png\" /> <jats:tex-math> $b_1(x) = x - 1/2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline8.png\" /> <jats:tex-math> $({-}1)^{n-1} b_n(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline9.png\" /> <jats:tex-math> $n gt 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline10.png\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-fold circular convolution of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000421_inline11.pn","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529133","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":"Forcing generalised quasirandom graphs efficiently","authors":"Andrzej Grzesik, Daniel Král’, Oleg Pikhurko","doi":"10.1017/s0963548323000263","DOIUrl":"https://doi.org/10.1017/s0963548323000263","url":null,"abstract":"\u0000 We study generalised quasirandom graphs whose vertex set consists of \u0000 \u0000 \u0000 \u0000$q$\u0000\u0000 \u0000 parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovász and Sós showed that the structure of such graphs is forced by homomorphism densities of graphs with at most \u0000 \u0000 \u0000 \u0000$(10q)^q+q$\u0000\u0000 \u0000 vertices; subsequently, Lovász refined the argument to show that graphs with \u0000 \u0000 \u0000 \u0000$4(2q+3)^8$\u0000\u0000 \u0000 vertices suffice. Our results imply that the structure of generalised quasirandom graphs with \u0000 \u0000 \u0000 \u0000$qge 2$\u0000\u0000 \u0000 parts is forced by homomorphism densities of graphs with at most \u0000 \u0000 \u0000 \u0000$4q^2-q$\u0000\u0000 \u0000 vertices, and, if vertices in distinct parts have distinct degrees, then \u0000 \u0000 \u0000 \u0000$2q+1$\u0000\u0000 \u0000 vertices suffice. The latter improves the bound of \u0000 \u0000 \u0000 \u0000$8q-4$\u0000\u0000 \u0000 due to Spencer.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91445411","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 the maximum number of edges in -critical graphs","authors":"Cong Luo, Jie Ma, Tianchi Yang","doi":"10.1017/s0963548323000238","DOIUrl":"https://doi.org/10.1017/s0963548323000238","url":null,"abstract":"<p>A graph is called <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000238:S0963548323000238_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-critical if its chromatic number is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000238:S0963548323000238_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span> but every proper subgraph has chromatic number less than <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000238:S0963548323000238_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>. An old and important problem in graph theory asks to determine the maximum number of edges in an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000238:S0963548323000238_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span>-vertex <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000238:S0963548323000238_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-critical graph. This is widely open for every integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000238:S0963548323000238_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$kgeq 4$</span></span></img></span></span>. Using a structural characterisation of Greenwell and Lovász and an extremal result of Simonovits, Stiebitz proved in 1987 that for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000238:S0963548323000238_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$kgeq 4$</span></span></img></span></span> and 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:S0963548323000238:S0963548323000238_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span>, this maximum number is less than the number of edges in the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231006153727984-0672:S0963548323000238:S0963548323000238_inline10.png\"><span data-mathjax-type=\"texmat","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529108","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}