$k$-loose elements and $k$-paving matroids

TL;DR

This paper analyzes $k$-loose elements and $k$-paving matroids, establishing sharp size and rank bounds that constrain how large such matroids can be given looseness conditions. It extends prior results from the $1$-loose case to general $k$, deriving a tight bound $|E(M)| \le 2^k(r-k+1)$ for simple binary matroids with a $k$-loose element and a GF$(q)$-matroid rank bound $r(M) \le (q+1)(k-1)+2q$ when two $k$-loose elements are present (unless they form a cocircuit). The paper further provides concrete consequences for $k$-paving matroids, including explicit rank and size bounds, and connects maximal binary $k$-paving matroids to Reed–Muller codes. Overall, the results deepen understanding of the structure and limitations of paving-like matroids and reveal ties to coding theory.

Abstract

For a matroid of rank $r$ and a non-negative integer $k$, an element is called $k$-loose if every circuit containing it has size greater than $r-k$. Zaslavsky and the author characterized all binary matroids with a $1$-loose element. In this paper, we establish a sharp linear bound on the size of a binary matroid, in terms of its rank, that contains a $k$-loose element. A matroid is called $k$-paving if all its elements are $k$-loose. Rajpal showed that for a prime power $q$, the rank of a $GF(q)$-matroid that is $k$-paving is bounded. We provide a bound on the rank of $GF(q)$-matroids that are cosimple and have two $k$-loose elements. Consequently, we deduce a bound on the rank of $GF(q)$-matroids that are $k$-paving. Additionally, we provide a bound on the size of binary matroids that are $k$-paving.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Proposition 1.4
• Theorem 1.5
• Corollary 1.6
• proof : Proof of Theorem \ref{['size_bound_one_k_loose']}
• Proposition 2.1
• proof
• Theorem 3.1