Clean tangled clutters, simplices, and projective geometries

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 $\tau=2$ Conjecture. Let $\mathcal{C}$ be a clean tangled clutter. It was recently proved that $\mathcal{C}$ has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the {\it core} of $\mathcal{C}$. The core is a duplication of the cuboid of a set of $0-1$ points, called the {\it setcore} of $\mathcal{C}$. 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 $\mathcal{C}$ 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.