The Circuit-Cocircuit Intersection Conjecture for Intersection Size $\le7$

TL;DR

This paper proves Oxley's conjecture that matroids with a circuit-cocircuit intersection of size up to 7 necessarily have a smaller intersection of size two less, confirming the conjecture for all such cases.

Contribution

The paper confirms Oxley's conjecture for intersection sizes up to 7, extending the known cases and providing new insights into matroid intersection properties.

Findings

Oxley's conjecture holds for intersection size ≤7

Matroids with a CCI of size k ≥4 have a CCI of size k-2 for k ≤7

Advances understanding of matroid intersection structure

Abstract

A circuit-cocircuit intersection, or a CCI for short, of a matroid is the intersection of a circuit and a cocircuit. Oxley conjectured (1992) that a matroid with a CCI of size $k\ge4$ has a CCI of size $k-2$. We show that the conjecture holds for $k\le7$.