The Two-Point Fano and Ideal Binary Clutters

Let F be a binary clutter. We prove that if F is non-ideal, then either F or its blocker b(F) has one of L7,O5,LC7 as a minor. L7 is the non-ideal clutter of the lines of the Fano plane, O5 is the non-ideal clutter of odd circuits of the complete graph K5, and the two-point Fano LC7 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 F is a minimally non-ideal binary clutter different from L7,O5, b(O5), then through every element, either F or b(F) has a two-point Fano minor.