Fernanda Rivera Omana

arXiv2026

Modularity, Extensions and Connectivity in Infinite Matroids

We generalize the well-studied notion of a modular pair of a finite matroid to arbitrary families of sets in infinite matroids, and use it to develop the theory of infinite matroids in several as-yet-unexplored areas. Our results include a complete theory of single-element extensions, a description of the relationship between quotients and projections, a proof that matroids for which every flat is modular must be finitary, and two new perspectives on the infinite matroid connectivity parameter λ. In most cases, existing theory for finite matroids either fails completely or does not extend in obvious ways, and as a result we develop multiple new techniques for reasoning about infinite matroids, including establishing well-behaved infinite analogues of nullity, local connectivity and skewness. We also point to an online repository containing formalized proofs of all our results using the lean4 proof assistant

math.CO↗

arXiv2023

On the density of matroids omitting a complete-graphic minor

We show that, if $M$ is a simple rank-$n$ matroid with no $\ell$-point line minor and no minor isomorphic to the cycle matroid of a $t$-vertex complete graph, then the ratio $|M| / n$ is bounded above by a singly exponential function of $\ell$ and $t$. We also bound this ratio in the special case where $M$ is a frame matroid, obtaining an answer that is within a factor of two of best-possible.

math.CO↗