{"title":"关于无限矩阵的打包/覆盖猜想","authors":"Attila Joó","doi":"10.1007/s11856-023-2595-4","DOIUrl":null,"url":null,"abstract":"<p>The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem of Edmonds and Fulkerson. A packing for a family <span>\\(({M_i}:i \\in \\Theta)\\)</span> of matroids on the common edge set <i>E</i> is a system <span>\\(({S_i}:i \\in \\Theta)\\)</span> of pairwise disjoint subsets of <i>E</i> where <i>S</i><sub><i>i</i></sub> is panning in <i>M</i><sub><i>i</i></sub>. Similarly, a covering is a system (<i>I</i><sub><i>i</i></sub>: <i>i</i> ∈ Θ) with <span>\\({\\cup _{i \\in \\Theta}}{I_i} = E\\)</span> where <i>I</i><sub><i>i</i></sub> is independent in <i>M</i><sub><i>i</i></sub>. The conjecture states that for every matroid family on <i>E</i> there is a partition <span>\\(E = {E_p} \\sqcup {E_c}\\)</span> such that <span>\\(({M_i}\\upharpoonright{E_p}:i \\in \\Theta)\\)</span> admits a packing and <span>\\(({M_i}.{E_c}:i \\in \\Theta)\\)</span> admits a covering. We prove the case where <i>E</i> is countable and each <i>M</i><sub><i>i</i></sub> is either finitary or cofinitary. To do so, we give a common generalisation of the singular matroid intersection theorem of Ghaderi and the countable case of the Matroid Intersection Conjecture by Nash-Williams by showing that the conjecture holds for countable matroids having only finitary and cofinitary components.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the packing/covering conjecture of infinite matroids\",\"authors\":\"Attila Joó\",\"doi\":\"10.1007/s11856-023-2595-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem of Edmonds and Fulkerson. A packing for a family <span>\\\\(({M_i}:i \\\\in \\\\Theta)\\\\)</span> of matroids on the common edge set <i>E</i> is a system <span>\\\\(({S_i}:i \\\\in \\\\Theta)\\\\)</span> of pairwise disjoint subsets of <i>E</i> where <i>S</i><sub><i>i</i></sub> is panning in <i>M</i><sub><i>i</i></sub>. Similarly, a covering is a system (<i>I</i><sub><i>i</i></sub>: <i>i</i> ∈ Θ) with <span>\\\\({\\\\cup _{i \\\\in \\\\Theta}}{I_i} = E\\\\)</span> where <i>I</i><sub><i>i</i></sub> is independent in <i>M</i><sub><i>i</i></sub>. The conjecture states that for every matroid family on <i>E</i> there is a partition <span>\\\\(E = {E_p} \\\\sqcup {E_c}\\\\)</span> such that <span>\\\\(({M_i}\\\\upharpoonright{E_p}:i \\\\in \\\\Theta)\\\\)</span> admits a packing and <span>\\\\(({M_i}.{E_c}:i \\\\in \\\\Theta)\\\\)</span> admits a covering. We prove the case where <i>E</i> is countable and each <i>M</i><sub><i>i</i></sub> is either finitary or cofinitary. To do so, we give a common generalisation of the singular matroid intersection theorem of Ghaderi and the countable case of the Matroid Intersection Conjecture by Nash-Williams by showing that the conjecture holds for countable matroids having only finitary and cofinitary components.</p>\",\"PeriodicalId\":14661,\"journal\":{\"name\":\"Israel Journal of Mathematics\",\"volume\":\"41 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-12-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Israel Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11856-023-2595-4\",\"RegionNum\":2,\"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":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-023-2595-4","RegionNum":2,"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
打包/覆盖猜想是鲍勒和卡梅辛在埃德蒙兹和富尔克森的矩阵分割定理的激励下提出的。共同边集 E 上的 Matroid 族 \(({M_i}:i \in \Theta)\)的打包是 E 的成对互不相交的子集系统 \(({S_i}:i \in \Theta)\),其中 Si 平移到 Mi 中。类似地,覆盖是一个具有 \({\cup _{i \in \Theta}}{I_i} = E\) 的系统(Ii: i∈ Θ),其中 Ii 在 Mi 中是独立的。这个猜想指出,对于 E 上的每个 matroid 族,都有一个分区 \(E = {E_p} \sqcup {E_c}\),使得 \(({M_i}\upharpoonright{E_p}:i \in \Theta)\) 允许打包,并且 \(({M_i}.{E_c}:i \in \Theta)\) 允许覆盖。我们将证明 E 是可数的且每个 Mi 要么是有限单元要么是共有限单元的情况。为此,我们给出了加达里(Ghaderi)的奇异矩阵交集定理和纳什-威廉姆斯(Nash-Williams)的矩阵交集猜想的可数情形的一般概括,证明了猜想对于只有有限元和共有限元成分的可数矩阵是成立的。
On the packing/covering conjecture of infinite matroids
The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem of Edmonds and Fulkerson. A packing for a family \(({M_i}:i \in \Theta)\) of matroids on the common edge set E is a system \(({S_i}:i \in \Theta)\) of pairwise disjoint subsets of E where Si is panning in Mi. Similarly, a covering is a system (Ii: i ∈ Θ) with \({\cup _{i \in \Theta}}{I_i} = E\) where Ii is independent in Mi. The conjecture states that for every matroid family on E there is a partition \(E = {E_p} \sqcup {E_c}\) such that \(({M_i}\upharpoonright{E_p}:i \in \Theta)\) admits a packing and \(({M_i}.{E_c}:i \in \Theta)\) admits a covering. We prove the case where E is countable and each Mi is either finitary or cofinitary. To do so, we give a common generalisation of the singular matroid intersection theorem of Ghaderi and the countable case of the Matroid Intersection Conjecture by Nash-Williams by showing that the conjecture holds for countable matroids having only finitary and cofinitary components.
期刊介绍:
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.