{"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}
引用次数: 0
Abstract
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.