Coloring games and algebraic problems on matroids

TL;DR

This thesis explores matroids and related structures like simplicial complexes and Euclidean spaces, addressing combinatorial, algebraic, and topological problems to deepen understanding of these fundamental combinatorial objects.

Contribution

It presents new results and insights into matroids and related structures, combining combinatorial, algebraic, and topological approaches.

Findings

New combinatorial properties of matroids

Algebraic characterizations of simplicial complexes

Topological insights into Euclidean space arrangements

Abstract

This thesis is basically devoted to matroids -- fundamental structure of combinatorial optimization -- though some of our results concern simplicial complexes, or Euclidean spaces. We study old and new problems for these structures, with combinatorial, algebraic, and topological flavor.