The Almost Intersection Property for Pairs of Matroids on Common Groundset

Bowler, Nathan; Carmesin, Johannes; Wojciechowski, Jerzy; Ghaderi, Shadisadat

doi:10.37236/8881

Cited by 3 publications

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“…The latter class consists of exactly those matroids that have only finite circuits which matroids are called finitary. Although several partial results have been obtained about the Matroid Intersection Conjecture (see [1], [10], [11], [12], [13], [14]), even the original finitary version is remained wide open.…”

Section: Introductionmentioning

confidence: 99%

Intersection of a partitional and a general infinite matroid

Joó¹

2020

Preprint

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Let E be a possibly infinite set and let M and N be matroids defined on E. We say that the pair {M, N } has the Intersection property if M and N share an independent set I admitting a bipartitionThe Matroid Intersection Conjecture of Nash-Williams says that every matroid pair has the Intersection property.It was shown by N. Bowler and J. Carmesin that the conjecture can be reduced to the special case where one of the matroids is a partitional matroid. We prove that if M is an arbitrary matroid and N is a partitional matroid of finitely many components, then {M, N } has the Intersection property.

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Section: Introductionmentioning

confidence: 99%

Intersection of a partitional and a general infinite matroid

Joó¹

2020

Preprint

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“…The content of this dissertation will be published in two papers: [14] and [23]. The content of [14] is described in Chapter 3 and the content of [23] is described in Chapters 4 and 5.…”

Section: Resultsmentioning

confidence: 99%

On the Matroid Intersection Conjecture

Ghaderi¹

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“…For instance it becomes clear that the class of patchwork matroids is closed under duality and taking minors. Furthermore, with Bowler et al's results in [10] it becomes immediately clear that the Matroid Intersection Conjecture holds whenever one of the matroids is patchwork. We finish Section 4 by showing the following two results Proposition 1.6.…”

mentioning

confidence: 89%

“…Aharoni and Ziv showed in [1] that the conjecture holds if one matroid is finitary and the other is a direct sum of finite rank matroids. Furthermore, Bowler et al show in [10] that it is sufficient for one of M and N to be patchwork.…”

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confidence: 99%

Infinite Matroids and Transfinite Sequences

Storm¹

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Infinite Matroids and Transfinite Sequences Martin Storm A matroid is a pair M = (E, I) where E is a set and I is a set of subsets of E that are called independent, echoing the notion of linear independence. One of the leading open problems in infinite matroid theory is the Matroid Intersection Conjecture by Nash-Williams which is a generalization of Hall's Theorem. In [31] Jerzy Wojciechowski introduced µ-admissibility for pairs of matroids on the same ground set and showed that it is a necessary condition for the existence of a matching. A pair of matroids (M, W) with common ground set E is µ-admissible if a subset of sequences in E × {0, 1} have a certain property. In order to determine if this property implies anything about the length of the sequence, we modify µ-admissibility to obtain µ-admissibility, an equivalent property for pairs of matroids. We then use µ-admissibility to show that for every successor ordinal of the form α + 2n there is a pair of partition matroids such that a shortest sequence in E × {0, 1} that fails to have the desired property has length α + 2n. Furthermore, we introduce the class of patchwork matroids, which contains all finite matroids and all uniform matroids, and provide a method for their construction, prove a characterization theorem, show the class is closed under duality and taking minors, as well as several other properties. Lastly, a cyclic flat is a flat which is a union of circuits. We combine this notion with that of trees of matroids, which are introduced in [7] to show that the lattice of cyclic flats of a locally finite tree of finite matroids contains an atomic element.

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The Almost Intersection Property for Pairs of Matroids on Common Groundset

Cited by 3 publications

References 5 publications

Intersection of a partitional and a general infinite matroid

Intersection of a partitional and a general infinite matroid

On the Matroid Intersection Conjecture

Infinite Matroids and Transfinite Sequences

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