Clean tangled clutters, simplices, and projective geometries

Abdi, Ahmad ORCID: 0000-0002-3008-4167, Cornuéjols, Gérard and Superdock, Matt (2022) Clean tangled clutters, simplices, and projective geometries. Journal of Combinatorial Theory, Series B, 154. 60 - 92. ISSN 0095-8956

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Abstract

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ = 2 Conjecture. Let C be a clean tangled clutter. It was recently proved that C has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of C. The core is a duplication of the cuboid of a set of 0 1 points, called the setcore of C. In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then C has the clutter of the lines of the Fano plane as a minor. Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.

Item Type:	Article
Official URL:	https://www.sciencedirect.com/journal/journal-of-c...
Additional Information:	Funding: Office of Naval Research (US) grant 00014-18-12129 and Natural Sciences and Engineering Research Council of Canada PDF grant 516584-2018. © 2021 Elsevier Inc.
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	21 Dec 2021 14:30
Last Modified:	20 Dec 2024 00:42
URI:	http://eprints.lse.ac.uk/id/eprint/113011

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