Maximal matroids in weak order posets

Abstract

Let $X$ be a family of subsets of a finite set E. A matroid on E is called an $X$-matroid if each set in $X$ is a circuit. We develop techniques for determining when there exists a unique maximal $X$-matroid in the weak order poset of all $X$-matroids on E and formulate a conjecture which would characterise the rank function of this unique maximal matroid when it exists. The conjecture suggests a new type of matroid rank function which extends the concept of weakly saturated sequences from extremal graph theory. We verify the conjecture for various families $X$ and show that, if true, the conjecture could have important applications in such areas as combinatorial rigidity and low rank matrix completion.