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

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.

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