The critical number of $I_{1,t}$-free triangle-free binary matroids

TL;DR

This paper proves that simple binary matroids that are both triangle-free and avoid certain intersections with rank-$t$ flats have a bounded critical number, revealing structural limitations in such matroids.

Contribution

The paper establishes that $I_{1,t}$-free and triangle-free binary matroids have a bounded critical number for all $t \,\geq\, 1$, a new structural result.

Findings

Bounded critical number for $I_{1,t}$-free triangle-free binary matroids.

Extension of previous results to all $t \geq 1$.

Structural limitations identified in binary matroids.

Abstract

A simple binary matroid, viewed as a restriction of a finite binary projective geometry $PG(n-1,2)$, is $I_{1,t}$-free if for any rank-$t$ flat of $PG(n-1,2)$, its intersection with the matroid is not a one-element set. In this paper, we show that the simple $I_{1,t}$-free and triangle-free binary matroids have bounded critical number for any $t \geq 1$.