On Functions of K-Balanced Matroids

In this paper, we prove an analogous to a result of Erdös and Rényi and of Kelly and Oxley. We also show that there are several properties of k-balanced matroids for which there exists a threshold function.


Introduction
We begin with some background material, which follows the terminology and notation in [1].Let ( ) , M E F = 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 [2]) the number r n       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 , r n U where 0,1, , r n =  .When r = n, the matroid , r r U is called free and when r = n = 0, the matroid 0,0 U is called the empty matroid.For more on matroid theory, the reader is referred to [1]- [15].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 ma- for all non-empty submatroids H M  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 r ω of the projective geometry ( ) by deleting elements so that each element has, independently of all other elements, probability 1 − 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 ( ) , r p P A denotes the probability that r ω 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 [16] [17].

K-Balanced Matroids
In this section, we prove the following main result which is analogous to Theorem 1 of Erdös and Rényi [16] and to Theorem 1.1 of Kelly and Oxley [17].Theorem 1.Let m and n be fixed positive integers with n ≤ m and suppose that , n m B 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 r ω has a submatroid isomorphic to some member of  for which the restriction ( ) where i ∝ equals the number of ordered pairs ( ) .

∑
We now want to obtain upper bounds on the numbers β is the number of ways to choose 1 A has been chosen, there are at most ( ) A must be contained in some rank n subspace W of PG(r-1,q) which contain the chosen set 1 2 A A  .The number δ of such subspaces W is bounded above by . But it was shown above that ( ) .
Now as we have by Equation (2), that ( ) Hence, by Equation ( 2), ( ) ( ) 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 needs to be k-balanced.For a more thorough discussion of this material, the reader is referred to Proposition 2 and Theorem 5 in [2].
We need to show that, in this case, be the set of subsets A of ( ) 1, PG r q −

1 Corollary 3
If n is a fixed positive integer, then a threshold function for the property that r ω has an n-element independent set is r q − .Corollary 2 If m is a fixed positive integer exceeding two, then a threshold function for the property that r ω has an m-element circuit is If n is a fixed positive integer, then a threshold function for the property that r ω contains a submatroid isomorphic to one to verify that an m-element circuit is k-balanced; this is precisely the uniform matroid 1, m m U − , while in Corollary 3, the projective geometry Let X and