Clean tangled clutters, simplices, and projective geometries
Ahmad Abdi, G\'erard Cornu\'ejols, Matt Superdock

TL;DR
This paper explores the geometric structure of the core of clean tangled clutters, revealing conditions under which it forms a simplex and its relation to projective geometries and the Fano plane.
Contribution
It establishes a connection between the setcore's convex hull being a simplex and the setcore being a cocycle space of a projective geometry, and links the simplex dimension to minors of the Fano plane.
Findings
The convex hull of the setcore is a full-dimensional polytope containing the hypercube center.
The polytope is a simplex iff the setcore is a projective geometry cocycle space.
High-dimensional simplices imply the presence of the Fano plane as a minor.
Abstract
A clutter is \emph{clean} if it has no delta or the blocker of an extended odd hole minor, and it is \emph{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 Conjecture. Let be a clean tangled clutter. It was recently proved that has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the {\it core} of . The core is a duplication of the cuboid of a set of points, called the {\it setcore} of . 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…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Coding theory and cryptography · graph theory and CDMA systems
