Extension to matroidal colorings

Determine whether the methods based on Gale’s HEX Lemma can be extended to prove colorful Helly-type overlap theorems for general matroidal colorings.

Background

The paper proves an overlap generalization of the colorful Helly theorem using an elementary argument based on Gale’s HEX Lemma. The authors contrast this approach with the algebraic methods of Kalai and Meshulam, which extend to matroidal colorings, and explicitly identify the extension of their own methods to that setting as unresolved.