The Circuit-Cocircuit Intersection Conjecture for Intersection Size $k\le6$

TL;DR

This paper proves Oxley's conjecture for matroids with circuit-cocircuit intersections of size up to 6, confirming that larger intersections imply smaller ones in these cases.

Contribution

The paper establishes the validity of Oxley's conjecture for all intersection sizes up to 6, extending previous partial results.

Findings

Oxley's conjecture holds for k ≤ 6

Matroids with large circuit-cocircuit intersections have smaller ones

The result confirms the conjecture for a new range of k

Abstract

Oxley conjectured (1992) that if a matroid has a circuit-cocircuit intersection of size $k\ge4$, it has a circuit-cocircuit intersection of size $k-2$. We show that this conjecture holds for $k\le6$.