Pub Date : 2023-04-20DOI: 10.1017/s0963548323000056
Andrew Newman
Abstract For a $k$ -uniform hypergraph $mathcal{H}$ on vertex set ${1, ldots, n}$ we associate a particular signed incidence matrix $M(mathcal{H})$ over the integers. For $mathcal{H} sim mathcal{H}_k(n, p)$ an Erdős–Rényi random $k$ -uniform hypergraph, ${mathrm{coker}}(M(mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = omega (1/n^{k - 1})$ , ${mathrm{coker}}(M(mathcal{H}))$ is torsion-free.
{"title":"Abelian groups from random hypergraphs","authors":"Andrew Newman","doi":"10.1017/s0963548323000056","DOIUrl":"https://doi.org/10.1017/s0963548323000056","url":null,"abstract":"Abstract For a $k$ -uniform hypergraph $mathcal{H}$ on vertex set ${1, ldots, n}$ we associate a particular signed incidence matrix $M(mathcal{H})$ over the integers. For $mathcal{H} sim mathcal{H}_k(n, p)$ an Erdős–Rényi random $k$ -uniform hypergraph, ${mathrm{coker}}(M(mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = omega (1/n^{k - 1})$ , ${mathrm{coker}}(M(mathcal{H}))$ is torsion-free.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135568825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-17DOI: 10.1017/s0963548323000032
Maria Axenovich, Christian Winter
Abstract Given partially ordered sets (posets) $(P, leq _P!)$ and $(P^{prime}, leq _{P^{prime}}!)$ , we say that $P^{prime}$ contains a copy of $P$ if for some injective function $f,:, Prightarrow P^{prime}$ and for any $X, Yin P$ , $Xleq _P Y$ if and only if $f(X)leq _{P^{prime}} f(Y)$ . For any posets $P$ and $Q$ , the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$ -dimensional Boolean lattice are coloured in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on a poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$ -dimensional Boolean lattice $Q_n$ , as $n$ grows large. We show a sharp jump in behaviour of this number as a function of $n$ depending on whether or not $P$ contains a copy of either a poset $V$ , that is a poset on elements $A, B, C$ such that $Bgt C$ , $Agt C$ , and $A$ and $B$ incomparable, or a poset $Lambda$ , its symmetric counterpart. Specifically, we prove that if $P$ contains a copy of $V$ or $Lambda$ then $R(P, Q_n) geq n +frac{1}{15} frac{n}{log n}$ . Otherwise $R(P, Q_n) leq n + c(P)$ for a constant $c(P)$ . This gives the first non-marginal improvement of a lower bound on poset Ramsey numbers and as a consequence gives $R(Q_2, Q_n) = n + Theta left(frac{n}{log n}right)$ .
{"title":"Poset Ramsey numbers: large Boolean lattice versus a fixed poset","authors":"Maria Axenovich, Christian Winter","doi":"10.1017/s0963548323000032","DOIUrl":"https://doi.org/10.1017/s0963548323000032","url":null,"abstract":"Abstract Given partially ordered sets (posets) $(P, leq _P!)$ and $(P^{prime}, leq _{P^{prime}}!)$ , we say that $P^{prime}$ contains a copy of $P$ if for some injective function $f,:, Prightarrow P^{prime}$ and for any $X, Yin P$ , $Xleq _P Y$ if and only if $f(X)leq _{P^{prime}} f(Y)$ . For any posets $P$ and $Q$ , the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$ -dimensional Boolean lattice are coloured in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on a poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$ -dimensional Boolean lattice $Q_n$ , as $n$ grows large. We show a sharp jump in behaviour of this number as a function of $n$ depending on whether or not $P$ contains a copy of either a poset $V$ , that is a poset on elements $A, B, C$ such that $Bgt C$ , $Agt C$ , and $A$ and $B$ incomparable, or a poset $Lambda$ , its symmetric counterpart. Specifically, we prove that if $P$ contains a copy of $V$ or $Lambda$ then $R(P, Q_n) geq n +frac{1}{15} frac{n}{log n}$ . Otherwise $R(P, Q_n) leq n + c(P)$ for a constant $c(P)$ . This gives the first non-marginal improvement of a lower bound on poset Ramsey numbers and as a consequence gives $R(Q_2, Q_n) = n + Theta left(frac{n}{log n}right)$ .","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135340270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-15DOI: 10.1017/s0963548322000372
Nicolás Rivera, Thomas Sauerwald, John Sylvester
Abstract Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $Omega ((n/k) log n)$ on the stationary cover time, holding for any $n$ -vertex graph $G$ and any $1 leq k =o(nlog n )$ . Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time . Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.
图上的随机游走是许多随机算法和随机过程的基本基元。人们很自然地会问，通过独立并行地运行$k$多个随机漫步可以获得多少收益。尽管多次行走的覆盖时间已经在许多自然网络中进行了研究，但寻找最坏情况起始点的多个覆盖时间的一般特征(由Alon, Avin, Koucký， Kozma, Lotker和Tuttle在2008年提出)仍然是一个开放的问题。首先，我们改进和收紧了$k$随机漫步从平稳分布中采样的顶点开始时的平稳覆盖时间的各种界限。例如，我们证明了平稳覆盖时间上$Omega ((n/k) log n)$的无条件下界，适用于任何$n$ -顶点图$G$和任何$1 leq k =o(nlog n )$。其次，我们在几个基本网络上建立了固定因子的多次行走的平稳覆盖时间。第三，我们提出了一个框架，用固定覆盖时间来描述最坏情况的覆盖时间，并提出了一个新的、宽松的多次行走混合时间概念，称为部分混合时间。粗略地说，部分混合时间只需要混合所有随机游走的特定部分。使用这些新概念，我们可以建立(或恢复)许多网络的最坏情况覆盖时间，包括扩展器、优先连接图、网格、二叉树和超立方体。
{"title":"Multiple random walks on graphs: mixing few to cover many","authors":"Nicolás Rivera, Thomas Sauerwald, John Sylvester","doi":"10.1017/s0963548322000372","DOIUrl":"https://doi.org/10.1017/s0963548322000372","url":null,"abstract":"Abstract Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $Omega ((n/k) log n)$ on the stationary cover time, holding for any $n$ -vertex graph $G$ and any $1 leq k =o(nlog n )$ . Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time . Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"260 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135582542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-14DOI: 10.1017/s0963548323000020
Richard Arratia, E. Rodney Canfield, Alfred W. Hales
Abstract For a random binary noncoalescing feedback shift register of width $n$ , with all $2^{2^{n-1}}$ possible feedback functions $f$ equally likely, the process of long cycle lengths, scaled by dividing by $N=2^n$ , converges in distribution to the same Poisson–Dirichlet limit as holds for random permutations in $mathcal{S}_N$ , with all $N!$ possible permutations equally likely. Such behaviour was conjectured by Golomb, Welch and Goldstein in 1959.
{"title":"Random feedback shift registers and the limit distribution for largest cycle lengths","authors":"Richard Arratia, E. Rodney Canfield, Alfred W. Hales","doi":"10.1017/s0963548323000020","DOIUrl":"https://doi.org/10.1017/s0963548323000020","url":null,"abstract":"Abstract For a random binary noncoalescing feedback shift register of width $n$ , with all $2^{2^{n-1}}$ possible feedback functions $f$ equally likely, the process of long cycle lengths, scaled by dividing by $N=2^n$ , converges in distribution to the same Poisson–Dirichlet limit as holds for random permutations in $mathcal{S}_N$ , with all $N!$ possible permutations equally likely. Such behaviour was conjectured by Golomb, Welch and Goldstein in 1959.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135727870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-03DOI: 10.1017/s0963548323000019
Tao Jiang, Sean Longbrake, Jie Ma
Abstract Given a family $mathcal{F}$ of bipartite graphs, the Zarankiewicz number $z(m,n,mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $mathcal{F}$ as a subgraph (such $G$ is called $mathcal{F}$ -free). For $1leq beta lt alpha lt 2$ , a family $mathcal{F}$ of bipartite graphs is $(alpha,beta )$ -smooth if for some $rho gt 0$ and every $mleq n$ , $z(m,n,mathcal{F})=rho m n^{alpha -1}+O(n^beta )$ . Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any $(alpha,beta )$ -smooth family $mathcal{F}$ , there exists $k_0$ such that for all odd $kgeq k_0$ and sufficiently large $n$ , any $n$ -vertex $mathcal{F}cup {C_k}$ -free graph with minimum degree at least $rho (frac{2n}{5}+o(n))^{alpha -1}$ is bipartite. In this paper, we strengthen their result by showing that for every real $delta gt 0$ , there exists $k_0$ such that for all odd $kgeq k_0$ and sufficiently large $n$ , any $n$ -vertex $mathcal{F}cup {C_k}$ -free graph with minimum degree at least $delta n^{alpha -1}$ is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families $mathcal{F}$ consisting of the single graph $K_{s,t}$ when $tgg s$ . We also prove an analogous result for $C_{2ell }$ -free graphs for every $ell geq 2$ , which complements a result of Keevash, Sudakov and Verstraëte.
{"title":"Bipartite-ness under smooth conditions","authors":"Tao Jiang, Sean Longbrake, Jie Ma","doi":"10.1017/s0963548323000019","DOIUrl":"https://doi.org/10.1017/s0963548323000019","url":null,"abstract":"Abstract Given a family \u0000$mathcal{F}$\u0000 of bipartite graphs, the Zarankiewicz number \u0000$z(m,n,mathcal{F})$\u0000 is the maximum number of edges in an \u0000$m$\u0000 by \u0000$n$\u0000 bipartite graph \u0000$G$\u0000 that does not contain any member of \u0000$mathcal{F}$\u0000 as a subgraph (such \u0000$G$\u0000 is called \u0000$mathcal{F}$\u0000 -free). For \u0000$1leq beta lt alpha lt 2$\u0000 , a family \u0000$mathcal{F}$\u0000 of bipartite graphs is \u0000$(alpha,beta )$\u0000 -smooth if for some \u0000$rho gt 0$\u0000 and every \u0000$mleq n$\u0000 , \u0000$z(m,n,mathcal{F})=rho m n^{alpha -1}+O(n^beta )$\u0000 . Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any \u0000$(alpha,beta )$\u0000 -smooth family \u0000$mathcal{F}$\u0000 , there exists \u0000$k_0$\u0000 such that for all odd \u0000$kgeq k_0$\u0000 and sufficiently large \u0000$n$\u0000 , any \u0000$n$\u0000 -vertex \u0000$mathcal{F}cup {C_k}$\u0000 -free graph with minimum degree at least \u0000$rho (frac{2n}{5}+o(n))^{alpha -1}$\u0000 is bipartite. In this paper, we strengthen their result by showing that for every real \u0000$delta gt 0$\u0000 , there exists \u0000$k_0$\u0000 such that for all odd \u0000$kgeq k_0$\u0000 and sufficiently large \u0000$n$\u0000 , any \u0000$n$\u0000 -vertex \u0000$mathcal{F}cup {C_k}$\u0000 -free graph with minimum degree at least \u0000$delta n^{alpha -1}$\u0000 is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families \u0000$mathcal{F}$\u0000 consisting of the single graph \u0000$K_{s,t}$\u0000 when \u0000$tgg s$\u0000 . We also prove an analogous result for \u0000$C_{2ell }$\u0000 -free graphs for every \u0000$ell geq 2$\u0000 , which complements a result of Keevash, Sudakov and Verstraëte.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135206022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-09DOI: 10.1017/s0963548322000360
David Conlon, Jacob Fox, Yuval Wigderson
Abstract The book graph $B_n ^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ . Here we consider the natural off-diagonal variant $r(B_{cn} ^{(k)}, B_n^{(k)})$ for fixed $c in (0,1]$ . In this more general setting, we show that an interesting dichotomy emerges: for very small $c$ , a simple $k$ -partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being $k$ -partite, while, for $c$ bounded away from $0$ , random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of $c$ .
{"title":"Off-diagonal book Ramsey numbers","authors":"David Conlon, Jacob Fox, Yuval Wigderson","doi":"10.1017/s0963548322000360","DOIUrl":"https://doi.org/10.1017/s0963548322000360","url":null,"abstract":"Abstract The book graph $B_n ^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ . Here we consider the natural off-diagonal variant $r(B_{cn} ^{(k)}, B_n^{(k)})$ for fixed $c in (0,1]$ . In this more general setting, we show that an interesting dichotomy emerges: for very small $c$ , a simple $k$ -partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being $k$ -partite, while, for $c$ bounded away from $0$ , random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of $c$ .","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136377500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-11DOI: 10.1017/S0963548322000256
Asaf Ferber, A. Sah, Mehtaab Sawhney, Yizhe Zhu
<jats:p>Motivated by problems from compressed sensing, we determine the threshold behaviour of a random <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000256_inline1.png" /> <jats:tex-math> $ntimes d pm 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> matrix <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000256_inline2.png" /> <jats:tex-math> $M_{n,d}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with respect to the property ‘every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000256_inline3.png" /> <jats:tex-math> $s$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> columns are linearly independent’. In particular, we show that for every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000256_inline4.png" /> <jats:tex-math> $0lt delta lt 1$ </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="S0963548322000256_inline5.png" /> <jats:tex-math> $s=(1-delta )n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000256_inline6.png" /> <jats:tex-math> $dleq n^{1+1/2(1-delta )-o(1)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> then with high probability every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000256_inline7.png" /> <jats:tex-math> $s$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> columns of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000256_inline8.png" /> <jats:tex-math> $M_{n,d}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are linearly independent, and if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000256_inline9.png" /> <jats:tex-math> $dgeq n^{1+1/2(1-delta )+o(1)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> then with high probability there are some <jats:inline-formula> <jats:alternatives> <jats:inline-graph
{"title":"Sparse recovery properties of discrete random matrices","authors":"Asaf Ferber, A. Sah, Mehtaab Sawhney, Yizhe Zhu","doi":"10.1017/S0963548322000256","DOIUrl":"https://doi.org/10.1017/S0963548322000256","url":null,"abstract":"\u0000\t <jats:p>Motivated by problems from compressed sensing, we determine the threshold behaviour of a random <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=\"S0963548322000256_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$ntimes d pm 1$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> matrix <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=\"S0963548322000256_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$M_{n,d}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> with respect to the property ‘every <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=\"S0963548322000256_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$s$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> columns are linearly independent’. In particular, we show that for every <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=\"S0963548322000256_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$0lt delta lt 1$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> 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=\"S0963548322000256_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$s=(1-delta )n$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, if <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=\"S0963548322000256_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$dleq n^{1+1/2(1-delta )-o(1)}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> then with high probability every <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=\"S0963548322000256_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$s$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> columns 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=\"S0963548322000256_inline8.png\" />\u0000\t\t<jats:tex-math>\u0000$M_{n,d}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> are linearly independent, and if <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=\"S0963548322000256_inline9.png\" />\u0000\t\t<jats:tex-math>\u0000$dgeq n^{1+1/2(1-delta )+o(1)}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> then with high probability there are some <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graph","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"5 1","pages":"316-325"},"PeriodicalIF":0.9,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86934360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-05DOI: 10.1017/s0963548322000268
R. Steiner
Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$ -minor is properly $(t-1)$ -colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$ -minor admits a vertex $(2t-2)$ -colouring such that all monochromatic components have size at most $lceil frac{1}{2}(t-2) rceil$ . The bound on the number of colours is optimal up to a factor of $2$ , improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput.17 815–821), Kang and Oum (2019, Combin. Probab. Comput.28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc.98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$ -minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$ -minor-free graphs with some additional ideas, which make the extension to odd $K_t$ -minor-free graphs possible.
为了加强Hadwiger的猜想，Gerards和Seymour在1995年推测，每个没有奇数$K_t$ -次的图都是适当的$(t-1)$ -可着色的。这就是著名的Odd Hadwiger猜想。我们证明了上述猜想的一个松弛性，即我们证明了每一个没有奇数K_t -次元的图都允许顶点$(2t-2)$ -着色，使得所有单色分量的大小最多为$lceil frc {1}{2}(t-2) rceil$。颜色数量的界限是最优的，高达2的因子，改进了Kawarabayashi (2008, Combin)对相同问题的先前界限。Probab。李志强，李志强(2019，北京)。Probab。刘和Wood (2021, arXiv预印本，arXiv:1905.09495)，并加强了van den Heuvel和Wood (2018, J. Lond.)的结果。数学。(Soc.98 129-148)，他证明了在更严格的假设下，即图是$K_t$ -次要的，上述结论成立。此外，我们的结果中组件大小的边界比以前的结果小得多，其中对$t$的依赖是由Robertson和Seymour的图小结构定理衍生的函数给出的。我们的简短证明结合了van den Heuvel和Wood对于无K_t次元图的方法和一些附加的思想，这使得扩展到奇数无K_t次元图成为可能。
{"title":"Improved bound for improper colourings of graphs with no odd clique minor","authors":"R. Steiner","doi":"10.1017/s0963548322000268","DOIUrl":"https://doi.org/10.1017/s0963548322000268","url":null,"abstract":"\u0000 Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd \u0000 \u0000 \u0000 \u0000$K_t$\u0000\u0000 \u0000 -minor is properly \u0000 \u0000 \u0000 \u0000$(t-1)$\u0000\u0000 \u0000 -colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd \u0000 \u0000 \u0000 \u0000$K_t$\u0000\u0000 \u0000 -minor admits a vertex \u0000 \u0000 \u0000 \u0000$(2t-2)$\u0000\u0000 \u0000 -colouring such that all monochromatic components have size at most \u0000 \u0000 \u0000 \u0000$lceil frac{1}{2}(t-2) rceil$\u0000\u0000 \u0000 . The bound on the number of colours is optimal up to a factor of \u0000 \u0000 \u0000 \u0000$2$\u0000\u0000 \u0000 , improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput.17 815–821), Kang and Oum (2019, Combin. Probab. Comput.28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc.98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is \u0000 \u0000 \u0000 \u0000$K_t$\u0000\u0000 \u0000 -minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on \u0000 \u0000 \u0000 \u0000$t$\u0000\u0000 \u0000 was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for \u0000 \u0000 \u0000 \u0000$K_t$\u0000\u0000 \u0000 -minor-free graphs with some additional ideas, which make the extension to odd \u0000 \u0000 \u0000 \u0000$K_t$\u0000\u0000 \u0000 -minor-free graphs possible.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"120 1","pages":"326-333"},"PeriodicalIF":0.9,"publicationDate":"2022-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74933860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1017/S096354832200013X
Xizhi Liu
A hypergraph F is non-trivial intersecting if every pair of edges in it have a nonempty intersection, but no vertex is contained in all edges of F . Mubayi and Verstraëte showed that for every k ≥ d + 1 ≥ 3 and n ≥ ( d + 1) k / d every k -graph H on n vertices without a non-trivial intersecting subgraph of size d + 1 contains at most (cid:2) n − 1 k − 1 (cid:3) edges. They conjectured that the same conclusion holds for all d ≥ k ≥ 4 and sufficiently large n . We confirm their conjecture by proving a stronger statement. They also conjectured that for m ≥ 4 and sufficiently large n the maximum size of a 3-graph on n vertices without a non-trivial intersecting subgraph of size 3 m + 1 is achieved by certain Steiner triple systems. We give a construction with more edges showing that their conjecture is not true in general.
{"title":"Hypergraphs without non-trivial intersecting subgraphs","authors":"Xizhi Liu","doi":"10.1017/S096354832200013X","DOIUrl":"https://doi.org/10.1017/S096354832200013X","url":null,"abstract":"A hypergraph F is non-trivial intersecting if every pair of edges in it have a nonempty intersection, but no vertex is contained in all edges of F . Mubayi and Verstraëte showed that for every k ≥ d + 1 ≥ 3 and n ≥ ( d + 1) k / d every k -graph H on n vertices without a non-trivial intersecting subgraph of size d + 1 contains at most (cid:2) n − 1 k − 1 (cid:3) edges. They conjectured that the same conclusion holds for all d ≥ k ≥ 4 and sufficiently large n . We confirm their conjecture by proving a stronger statement. They also conjectured that for m ≥ 4 and sufficiently large n the maximum size of a 3-graph on n vertices without a non-trivial intersecting subgraph of size 3 m + 1 is achieved by certain Steiner triple systems. We give a construction with more edges showing that their conjecture is not true in general.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"28 1","pages":"1076-1091"},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75445155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-29DOI: 10.1017/s0963548322000244
L. Mili'cevi'c
<jats:p>Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000244_inline1.png" /> <jats:tex-math> $V$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a finite-dimensional vector space over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000244_inline2.png" /> <jats:tex-math> $mathbb{F}_p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We say that a multilinear form <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000244_inline3.png" /> <jats:tex-math> $alpha colon V^k to mathbb{F}_p$ </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="S0963548322000244_inline4.png" /> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> variables is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000244_inline5.png" /> <jats:tex-math> $d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:italic>approximately symmetric</jats:italic> if the partition rank of difference <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000244_inline6.png" /> <jats:tex-math> $alpha (x_1, ldots, x_k) - alpha (x_{pi (1)}, ldots, x_{pi (k)})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is at most <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000244_inline7.png" /> <jats:tex-math> $d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for every permutation <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000244_inline8.png" /> <jats:tex-math> $pi in textrm{Sym}_k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In a work concerning the inverse theorem for the Gowers uniformity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000244_inline9.png" /> <jats:tex-math> $|!cdot! |_{mathsf{U}^4}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> norm in the case of low characteristic, Tidor conjectured that any <jats:inline-formula>
设$V$是$mathbb{F}_p$上的有限维向量空间。我们说$k$变量中的多元线性形式$alpha colon V^k to mathbb{F}_p$是$d$ -近似对称的，如果对于每个排列$pi in textrm{Sym}_k$，差分$alpha (x_1, ldots, x_k) - alpha (x_{pi (1)}, ldots, x_{pi (k)})$的划分秩最多为$d$。Tidor在一篇关于低特征情况下Gowers均匀性$|!cdot! |_{mathsf{U}^4}$范数的反定理的著作中，推测任何$d$ -近似对称的多线性形式$alpha colon V^k to mathbb{F}_p$与对称的多线性形式最多有一个分秩的多线性形式$O_{p,k,d}(1)$的区别，并在三线性形式的情况下证明了这一猜想。在本文中，有些令人惊讶的是，我们证明了这个猜想是错误的。事实上，通过构造一个3-近似对称的多元线性形式$alpha colon mathbb{F}_2^n times mathbb{F}_2^n times mathbb{F}_2^n times mathbb{F}_2^n to mathbb{F}_2$，我们证明了近似对称的形式可以与对称的形式相距甚远，而$alpha$与任何对称的多元线性形式的区别至少为$Omega (sqrt [3]{n})$。
{"title":"Approximately symmetric forms far from being exactly symmetric","authors":"L. Mili'cevi'c","doi":"10.1017/s0963548322000244","DOIUrl":"https://doi.org/10.1017/s0963548322000244","url":null,"abstract":"\u0000\t <jats:p>Let <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=\"S0963548322000244_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$V$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> be a finite-dimensional vector space over <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=\"S0963548322000244_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$mathbb{F}_p$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. We say that a multilinear form <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=\"S0963548322000244_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$alpha colon V^k to mathbb{F}_p$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> in <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=\"S0963548322000244_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$k$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> variables is <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=\"S0963548322000244_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$d$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-<jats:italic>approximately symmetric</jats:italic> if the partition rank of difference <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=\"S0963548322000244_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$alpha (x_1, ldots, x_k) - alpha (x_{pi (1)}, ldots, x_{pi (k)})$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is at most <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=\"S0963548322000244_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$d$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> for every permutation <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=\"S0963548322000244_inline8.png\" />\u0000\t\t<jats:tex-math>\u0000$pi in textrm{Sym}_k$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. In a work concerning the inverse theorem for the Gowers uniformity <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=\"S0963548322000244_inline9.png\" />\u0000\t\t<jats:tex-math>\u0000$|!cdot! |_{mathsf{U}^4}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> norm in the case of low characteristic, Tidor conjectured that any <jats:inline-formula>\u0000\t","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":"338 2-3","pages":"299-315"},"PeriodicalIF":0.9,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72492977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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