A random graph on n vertices is a random subgraph of the complete graph on n vertices. By analogy with this, the present paper studies the asymptotic properties of a random submatroid ωr of the projective geometry PG(r— 1, q). The main result concerns Kr, the rank of the largest projective geometry occurring as a submatroid of ωr. We show that with probability one, for sufficiently large r, Krtakes one of at most two values depending on r. This theorem is analogous to a result of Bollobás and

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... dös on the clique number of a random graph. However, whereas from the matroid theorem one can essentially determine the critical exponent of ωr the graph theorem gives only a lower bound on the chromatic number of a random graph.