The Two-Point Fano and Ideal Binary Clutters

Abstract

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 L 7 , O 5 , LC 7 as a minor. L 7 is the non-ideal clutter of the lines of the Fano plane, O 5 is the non-ideal clutter of odd circuits of the complete graph K 5 , and the two-point Fano LC 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 F is a minimally non-ideal binary clutter different from L 7 , O 5 , b(O 5), then through every element, either F or b(F) has a two-point Fano minor.

The Minimally Non-Ideal Binary Clutters with a Triangle

“…Then F/I \ J denotes the clutter over ground set E(F) − (I ∪ J) that consists of the minimal sets in {S − I : S ∈ F, S ∩ J = ∅}. 1 We say that F/I \ J is a minor of F; it is a proper minor if I ∪ J = ∅. It can be readily checked that if F is binary, then so are all its minors [18].…”

“…Date: August 15, 2018. This work is supported by NSERC CGS and Discovery grants and by U.S. Office of Naval Research grants under award numbers N00014-15-1-2171 and N00014-18-1-2078. 1 Given sets A, B we denote by A − B the set {a ∈ A : a / ∈ B} and, for element a, we write A − a instead of A − {a}. The weak f -flowing conjecture.…”

“…Before delving into the proof, let us say a few words about our result. In another paper [1], we use Theorem 1 to prove that if F is an mni binary clutter that is different from L 7 , O 5 , b(O 5 ), then through every element of E(F), one of F, b(F) has a "two-point Fano minor". This consequence provides yet another compelling evidence in support of the f -flowing conjecture.…”