#### Title

Asymptotic Properties of Random Subsets of Projective Spaces

#### Document Type

Article

#### Publication Date

1-1-1982

#### Abstract

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

#### First Page

119

#### Last Page

130

#### Recommended Citation

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