Extremal bounds for pattern avoidance in multidimensional 0-1 matrices

IF 0.7 3区 数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2024-10-31 DOI:10.1016/j.disc.2024.114303
{"title":"Extremal bounds for pattern avoidance in multidimensional 0-1 matrices","authors":"","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>&gt;</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":null,"pages":null},"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 P-saturated where P is a family of 0-1 matrices if M avoids every element of P and changing any 0-entry of M to a 1-entry introduces a copy of some element of P. The extremal function ex(n,P) and saturation function sat(n,P) are the maximum and minimum possible number of 1-entries in an n×n P-saturated 0-1 matrix, respectively, and the semisaturation function ssat(n,P) is the minimum possible number of 1-entries in an n×n P-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 P.
We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-O(nd1) 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-O(nd1) d-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers k,d and integer r[0,d1], we construct a family of d-dimensional 0-1 matrices with both extremal function and saturation function exactly knr for sufficiently large n. We show that no family of d-dimensional 0-1 matrices has saturation function strictly between O(1) and Θ(n) and we construct a family of d-dimensional 0-1 matrices with bounded saturation function and extremal function Ω(ndϵ) for any ϵ>0. 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 Θ(nr) for some integer r[0,d1].
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
多维 0-1 矩阵中模式规避的极值界限
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)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
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.
期刊最新文献
Extremal bounds for pattern avoidance in multidimensional 0-1 matrices Generation of 3-connected, planar line graphs A note on locating-dominating sets in twin-free graphs Light 3-faces in 3-polytopes without adjacent triangles Large matchings in maximal 1-planar graphs
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1