The Two-Point Fano and Ideal Binary Clutters

Abstract

Let ${\mathbb {F}}$ be a binary clutter. We prove that if ${\mathbb {F}}$ is non-ideal, then either ${\mathbb {F}}$ or its blocker $b({\mathbb {F}})$ has one of ${\mathbb {L}}_7,{\mathbb {O}}_5,{\mathbb {L}}{\mathbb {C}}_7$ as a minor. ${\mathbb {L}}_7$ is the non-ideal clutter of the lines of the Fano plane, ${\mathbb {O}}_5$ is the non-ideal clutter of odd circuits of the complete graph $K_5$, and the two-point Fano ${\mathbb {L}}{\mathbb {C}}_7$ is the ideal clutter whose sets are the lines, and their complements, of the Fano plane that contain exactly one of two fixed points. In fact, we prove the following stronger statement: if ${\mathbb {F}}$ is a minimally non-ideal binary clutter different from ${\mathbb {L}}_7,{\mathbb {O}}_5,b({\mathbb {O}}_5)$, then through every element, either ${\mathbb {F}}$ or $b({\mathbb {F}})$ has a two-point Fano minor.