{"title":"多维 0-1 矩阵中模式规避的极值界限","authors":"Jesse Geneson , Shen-Fu Tsai","doi":"10.1016/j.disc.2024.114303","DOIUrl":null,"url":null,"abstract":"<div><div>A 0-1 matrix <em>M</em> contains another 0-1 matrix <em>P</em> if some submatrix of <em>M</em> can be turned into <em>P</em> by changing any number of 1-entries to 0-entries. The 0-1 matrix <em>M</em> is <span><math><mi>P</mi></math></span>-saturated where <span><math><mi>P</mi></math></span> is a family of 0-1 matrices if <em>M</em> avoids every element of <span><math><mi>P</mi></math></span> and changing any 0-entry of <em>M</em> to a 1-entry introduces a copy of some element of <span><math><mi>P</mi></math></span>. The extremal function <span><math><mi>ex</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> and saturation function <span><math><mi>sat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> are the maximum and minimum possible number of 1-entries in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> <span><math><mi>P</mi></math></span>-saturated 0-1 matrix, respectively, and the semisaturation function <span><math><mi>ssat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> is the minimum possible number of 1-entries in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> <span><math><mi>P</mi></math></span>-semisaturated 0-1 matrix <em>M</em>, i.e., changing any 0-entry in <em>M</em> to a 1-entry introduces a new copy of some element of <span><math><mi>P</mi></math></span>.</div><div>We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-<span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> <em>d</em>-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-<span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> <em>d</em>-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers <span><math><mi>k</mi><mo>,</mo><mi>d</mi></math></span> and integer <span><math><mi>r</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>, we construct a family of <em>d</em>-dimensional 0-1 matrices with both extremal function and saturation function exactly <span><math><mi>k</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> for sufficiently large <em>n</em>. We show that no family of <em>d</em>-dimensional 0-1 matrices has saturation function strictly between <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mi>Θ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and we construct a family of <em>d</em>-dimensional 0-1 matrices with bounded saturation function and extremal function <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math></span> for any <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span>. Up to a constant multiplicative factor, we fully settle the problem of characterizing the semisaturation function of families of <em>d</em>-dimensional 0-1 matrices, which we prove to always be <span><math><mi>Θ</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></math></span> for some integer <span><math><mi>r</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114303"},"PeriodicalIF":0.7000,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Extremal bounds for pattern avoidance in multidimensional 0-1 matrices\",\"authors\":\"Jesse Geneson , Shen-Fu Tsai\",\"doi\":\"10.1016/j.disc.2024.114303\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>A 0-1 matrix <em>M</em> contains another 0-1 matrix <em>P</em> if some submatrix of <em>M</em> can be turned into <em>P</em> by changing any number of 1-entries to 0-entries. The 0-1 matrix <em>M</em> is <span><math><mi>P</mi></math></span>-saturated where <span><math><mi>P</mi></math></span> is a family of 0-1 matrices if <em>M</em> avoids every element of <span><math><mi>P</mi></math></span> and changing any 0-entry of <em>M</em> to a 1-entry introduces a copy of some element of <span><math><mi>P</mi></math></span>. The extremal function <span><math><mi>ex</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> and saturation function <span><math><mi>sat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> are the maximum and minimum possible number of 1-entries in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> <span><math><mi>P</mi></math></span>-saturated 0-1 matrix, respectively, and the semisaturation function <span><math><mi>ssat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> is the minimum possible number of 1-entries in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> <span><math><mi>P</mi></math></span>-semisaturated 0-1 matrix <em>M</em>, i.e., changing any 0-entry in <em>M</em> to a 1-entry introduces a new copy of some element of <span><math><mi>P</mi></math></span>.</div><div>We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-<span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> <em>d</em>-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-<span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> <em>d</em>-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers <span><math><mi>k</mi><mo>,</mo><mi>d</mi></math></span> and integer <span><math><mi>r</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>, we construct a family of <em>d</em>-dimensional 0-1 matrices with both extremal function and saturation function exactly <span><math><mi>k</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> for sufficiently large <em>n</em>. We show that no family of <em>d</em>-dimensional 0-1 matrices has saturation function strictly between <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mi>Θ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and we construct a family of <em>d</em>-dimensional 0-1 matrices with bounded saturation function and extremal function <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math></span> for any <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span>. Up to a constant multiplicative factor, we fully settle the problem of characterizing the semisaturation function of families of <em>d</em>-dimensional 0-1 matrices, which we prove to always be <span><math><mi>Θ</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></math></span> for some integer <span><math><mi>r</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>.</div></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":\"348 2\",\"pages\":\"Article 114303\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-10-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24004345\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004345","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
如果 M 的某个子矩阵可以通过将任意数量的 1 条目变为 0 条目而变成 P,则 0-1 矩阵 M 包含另一个 0-1 矩阵 P。如果 M 避开了 P 的每个元素,并且将 M 的任意 0 条目改为 1 条目都会引入 P 的某个元素的副本,那么 0-1 矩阵 M 就是 P 饱和的,其中 P 是 0-1 矩阵族。极值函数 ex(n,P) 和饱和函数 sat(n,P) 分别是 n×n P 饱和 0-1 矩阵中 1 条目的最大可能数目和最小可能数目,而半饱和函数 ssat(n,P) 是 n×n P 半饱和 0-1 矩阵 M 中 1 条目的最小可能数目,即、我们研究多维 0-1 矩阵的这些函数。特别是,我们给出了最小非 O(nd-1)d 维 0-1 矩阵参数的上限,这是从二维最小非线性 0-1 矩阵推广而来的;我们还证明了存在无限多的最小非 O(nd-1)d 维 0-1 矩阵,且所有维的长度都大于 1。对于任意正整数 k,d 和整数 r∈[0,d-1],我们构造了一个 d 维 0-1 矩阵族,其极值函数和饱和函数在足够大的 n 条件下正好为 knr。我们证明没有一个 d 维 0-1 矩阵族的饱和函数严格介于 O(1) 和 Θ(n) 之间,并且我们构造了一个 d 维 0-1 矩阵族,其饱和函数和极值函数 Ω(nd-ϵ) 对于任意 ϵ>0 都是有界的。对于某个整数 r∈[0,d-1],我们证明其半饱和函数总是 Θ(nr)。
Extremal bounds for pattern avoidance in multidimensional 0-1 matrices
A 0-1 matrix M contains another 0-1 matrix P if some submatrix of M can be turned into P by changing any number of 1-entries to 0-entries. The 0-1 matrix M is -saturated where is a family of 0-1 matrices if M avoids every element of and changing any 0-entry of M to a 1-entry introduces a copy of some element of . The extremal function and saturation function are the maximum and minimum possible number of 1-entries in an -saturated 0-1 matrix, respectively, and the semisaturation function is the minimum possible number of 1-entries in an -semisaturated 0-1 matrix M, i.e., changing any 0-entry in M to a 1-entry introduces a new copy of some element of .
We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non- d-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non- d-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers and integer , we construct a family of d-dimensional 0-1 matrices with both extremal function and saturation function exactly for sufficiently large n. We show that no family of d-dimensional 0-1 matrices has saturation function strictly between and and we construct a family of d-dimensional 0-1 matrices with bounded saturation function and extremal function for any . Up to a constant multiplicative factor, we fully settle the problem of characterizing the semisaturation function of families of d-dimensional 0-1 matrices, which we prove to always be for some integer .
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.