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 Erdös 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. © 1982, Cambridge Philosophical Society. All rights reserved.
Publication Source (Journal or Book title)
Mathematical Proceedings of the Cambridge Philosophical Society
Kelly, D., & Oxley, J. (1982). Asymptotic Properties of Random Subsets of Projective Spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 91 (1), 119-130. https://doi.org/10.1017/S0305004100059181