Idealness of $k$-wise intersecting families

TL;DR

This paper investigates a conjecture that for certain intersecting structures called $k$-wise intersecting clutters, with $k extgreater 3$, these are non-ideal, providing proofs for the case $k=4$ in binary clutters and discussing related concepts.

Contribution

The paper proves the conjecture for $k=4$ in binary clutters and explores connections to various combinatorial and graph-theoretic concepts.

Findings

Proved the conjecture for $k=4$ in binary clutters.

Connected the conjecture to Jaeger's 8-flow theorem and Seymour's matroid characterization.

Discussed implications for chromatic number, projective geometries, and cycle covers.

Abstract

A clutter is \emph{$k$-wise intersecting} if every $k$ members have a common element, yet no element belongs to all members. We conjecture that, for some integer $k\geq 4$, every $k$-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for $k=4$ for the class of binary clutters. Two key ingredients for our proof are Jaeger's $8$-flow theorem for graphs, and Seymour's characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters.