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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 intoclasses 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), whoshowed that $\text{PG}(n-1,q)$ is not $(1,c)$-decomposable, resp. not$(b,c)$-decomposable.

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