We begin with some background material, which follows the terminology and notation in  . Let denote the matroid on the ground set E with flats F. All matroids considered in this paper are loopless. In particular, if M is a matroid on a set E and X ⊆ E, then r(X) will denote the rank of X in M. We shall be considering projective geometries over a fixed finite field GF(q), recalling that
(see, for example  ) the number of rank-n subspaces of the projective
geometry PG (r − 1, q) is
The uniform matroid of rank r and size n is denoted by where
. When r = n, the matroid is called free and when r = n = 0, the matroid is called the empty matroid. For more on matroid theory, the reader is referred to  -  . Let k be a nonnegative integer. The k-density of a
matroid M with rank greater than k is given by , where |M|
is the size of the ground set of M and r(M) is the rank of the matroid M. A matroid M is k-balanced if and
for all non-empty submatroids and strictly k-balanced if the inequality is strict for all such H ≠ M. When k = 0, M is called balanced and when k = 1, M is called strongly balanced.
A random submatroid of the projective geometry is obtained from by deleting elements so that each element has, independently of all other elements, probability 1 − p of being deleted and probability 1 − p of being retained. In this paper, we take p to be a function p(r) of r. Let A be a fixed property which a matroid may or may not possess and denotes the probability that has property A. We shall show that there are several properties A of k-balanced matroids for which there exists a function t(r) such that
If such a function exists, it is called a threshold function for the property A. For more on these notions, the reader is referred   .
2. K-Balanced Matroids
In this section, we prove the following main result which is analogous to Theorem 1 of Erdös and Rényi  and to Theorem 1.1 of Kelly and Oxley  .
Theorem 1. Let m and n be fixed positive integers with n ≤ m and suppose that denote a non-empty set of k-balanced simple matroids, each of which have m elements and rank n and is representable over GF(q). Then a threshold function for the property B that has a submatroid isomorphic to some
member of is .
Proof. Let X and denote the number of submatroids of the matroid and respectively which are isomorphic to some member of . Then
by definition of expectation. Therefore
Thus, if , then .
Now suppose that . We need to show that, in this case, . Let be the set of subsets A of for which the
restriction of to A is isomorphic to some member of . Then
where equals the number of ordered pairs such that and . Thus
We now want to obtain upper bounds on the numbers , so suppose that and where . Then as is k-balanced,
and so . It follows that
and hence where is the floor of .
Now where is the number of ways to choose and is the number of ways to choose so that , having already
been chosen. Clearly . Once has been chosen, there are at
most choices for the subset of . Further, once has been chosen, must be contained in some rank n subspace W of PG(r-1,q) which contain the chosen set . The number δ of such subspaces W is bounded above by
where . Thus . But it was shown above that
; hence Once W has been chosen, there are at most choices for . We conclude that
Now as we have by Equation (2), that
Hence, by Equation (2),
Since . Thus
Now consider . We have
Thus . But hence for . It follows from Equation (4) that ; hence . Therefore, by Chebyshev’s Inequality, . We conclude that is indeed a threshold function for the property B.
Corollary 1 If n is a fixed positive integer, then a threshold function for the property that has an n-element independent set is .
Corollary 2 If m is a fixed positive integer exceeding two, then a threshold
function for the property that has an m-element circuit is .
Corollary 3 If n is a fixed positive integer, then a threshold function for the property that contains a submatroid isomorphic to is
To show that the preceding three results are valid, we are required to check that the appropriate submatroids are k-balanced. For example, in Corollary 1, the n-element independent set must be k-balanced; this is the free matroid . Corollary 2 requires one to verify that an m-element circuit is k-balanced; this is precisely the uniform matroid , while in Corollary 3, the projective geometry needs to be k-balanced. For a more thorough discussion of this material, the reader is referred to Proposition 2 and Theorem 5 in  .