Combinatorial representations

TL;DR

This paper introduces combinatorial representations that generalize linear matroid representations, showing their applicability to families of subsets and graphs, and exploring representations with limited matrix rows.

Contribution

It extends the concept of matroid representations to combinatorial forms, characterizes representability over various alphabets, and analyzes low-row matrix representations.

Findings

Any family of same-cardinality subsets has a combinatorial representation.

Graphs are representable over sufficiently large alphabets.

Characterization of families representable over specific alphabets.

Abstract

This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then prove that any graph is representable over all alphabets of size larger than some number depending on the graph. We also provide a characterisation of families representable over a given alphabet. Then, we associate a rank function and a rank operator to any representation which help us determine some criteria for the functions used in a representation. While linearly representable matroids can be viewed as having representations via matrices with only one row, we conclude this paper by an investigation of representations via matrices with only two rows.