Random GF(q)-representable matroids are not (b,c)-decomposable

TL;DR

This paper demonstrates that random subsets of projective geometries over finite fields are unlikely to be decomposable into small classes with certain coloring properties, extending previous non-decomposability results.

Contribution

It generalizes prior work by proving that random subsets of projective geometries are not $(b,c)$-decomposable for broader parameters, advancing understanding of their structural complexity.

Findings

Random subsets are not $(b,c)$-decomposable with high probability.

Extends previous non-decomposability results to broader parameters.

Shows limitations on partitioning and coloring properties of these subsets.

Abstract

We show that a random subset of the rank-$n$ projective geometry $\text{PG}(n-1,q)$ is, with high probability, not $(b,c)$-decomposable: if $k$ is its colouring number, it does not admit a partition of its ground set into classes of size at most $ck$, every transversal of which is $b$-colourable. This generalises recent results by Abdolazimi, Karlin, Klein, and Oveis Gharan (arXiv:2111.12436) and by Leichter, Moseley, and Pruhs (arXiv:2206.12896), who showed that $\text{PG}(n-1,q)$ is not $(1,c)$-decomposable, resp. not $(b,c)$-decomposable.