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

( q r − 1 ) ( q r − 1 − 1 ) ⋯ ( q r − n + 1 − 1 ) ( q n − 1 ) ( q n − 1 − 1 ) ⋯ ( q − 1 ) .

The uniform matroid of rank r and size n is denoted by U r , n where

r = 0 , 1 , ⋯ , n . When r = n, the matroid U r , r is called free and when r = n = 0, the matroid U 0 , 0 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 d k ( M ) = | M | r ( M ) − k , 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 r ( M ) > ( k ( k + 1 ) ) / 2 and

d k ( M ) ≤ d k ( M ) (1)

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 ω r of the projective geometry P G ( r − 1 , q ) is obtained from P G ( r − 1 , q ) 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 P r , p ( A ) denotes the probability that ω r 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

lim r → ∞ P r , p ( A ) = { 0 ,     lim r → ∞ P t ( r ) = 0 1 ,     lim r → ∞ P t ( r ) = ∞

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] .

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 B n , m 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 ω r has a submatroid isomorphic to some

member of B n , m is q − r n m .

Proof. Let X and B n , m denote the number of submatroids of the matroid ω r and P G ( n − 1 , q ) respectively which are isomorphic to some member of B n , m . Then

P r , p ( B ) = P ( X ≠ 0 ) ≤ E X

by definition of expectation. Therefore

P r , p ( B ) ≤ [ r n ] B n , m p m ≤ B n , m p m q r n ≤ B n , m ( p q − r n m ) m .

Thus, if lim r → ∞ p q − r n m = 0 , then lim r → ∞ P r , p ( B ) = 0 .

Now suppose that lim n → ∞ p q − r n m = ∞ . We need to show that, in this case, lim n → ∞ P r , p ( B ) = 1 . Let D m , n be the set of subsets A of P G ( r − 1 , q ) for which the

restriction P G ( r − 1 , q ) | A of P G ( r − 1 , q ) to A is isomorphic to some member of B n , m . Then

E X 2 = ∑ A 1 ∈ D m , n ∑ A 2 ∈ D m , n p | A 1 ∪ A 2 | = ∑ i = 0 m     p m + i ∝ i (2)

where ∝ i equals the number of ordered pairs ( A 1 , A 2 ) such that A 1 , A 2 ∈ D m , n and | A 1 ∩ A 2 | = m − i . Thus

E X 2 ≤ p 2 m [ ( B m , n [ r n ] ) 2 + ∑ i = 0 m − 1     p i − m ∝ i ] .

We now want to obtain upper bounds on the numbers ∝ 0 , ∝ 1 , ⋯ , ∝ m − 1 , so suppose that A 1 , A 2 ∈ D m , n and | A 1 ∩ A 2 | = m − i where 0 ≤ i ≤ m − 1 . Then as P G ( r − 1 , q ) | A is k-balanced,

( | A 1 ∩ A 2 | ) / ( r ( A 1 ∩ A 2 ) − k ) ≤ m / ( n − k )

and so r ( A 1 ∩ A 2 ) ≥ ( ( m − i ) ( n − k ) ) / m + k . It follows that

r ( A 2 ) − r ( A 1 ∩ A 2 ) ≤ n − ( ( m − i ) ( n − k ) ) / m − k = ( i ( n − k ) ) / m ≤ ( i n ) / m

and hence r ( A 2 ) − r ( A 1 ∩ A 2 ) ≤ ⌊ ( i n ) / m ⌋ where ⌊ ( i n ) / m ⌋ is the floor of ( i n ) / m .

Now ∝ i = β i γ i where β i is the number of ways to choose A 1 and γ i is the number of ways to choose A 2 so that | A 1 ∩ A 2 | = m − i , A 1 having already

been chosen. Clearly β i = B m , n [ r n ] . Once A 1 has been chosen, there are at

most ( m − i m ) choices for the subset A 1 ∩ A 2 of A 1 . Further, once A 1 ∩ A 2 has been chosen, A 2 must be contained in some rank n subspace W of PG(r-1,q) which contain the chosen set A 1 ∩ A 2 . The number δ of such subspaces W is bounded above by

( ( q r − q s ) / ( q − 1 ) ) ( ( q r − q s + 1 ) / ( q − 1 ) ) ⋯ ( ( q r − q n − 1 ) / ( q − 1 ) ) ,

where s = r ( A 1 ∩ A 2 ) . Thus δ ≤ q r ( n − 1 ) . But it was shown above that

n − s ≤ ⌊ ( i n ) / m ⌋ ; hence δ ≤ q r ⌊ i n / m ⌋ . Once W has been chosen, there are at most B m , n choices for A 2 . We conclude that

γ i ≤ ( m − i m ) q r ⌊ i n / m ⌋ B m , n

and hence

α i ≤ [ r n ] B m , n 2 ( m − i m ) q r ⌊ i n / m ⌋ . (3)

Now as E X = [ r n ] B m , n p m , we have by Equation (2), that

E X 2 ( E X ) 2 ≤ 1 + ( B m , n [ r n ] ) − 2 + ∑ i = 0 m − 1   p i − m ∝ i .

Hence, by Equation (2),

E X 2 ( E X ) 2 ≤ 1 + ( B m , n [ r n ] ) − 2 + ∑ i = 0 m − 1     p i − m [ r n ] B m , n 2 ( m − i m ) q r ⌊ i n m ⌋ . .

Thus E X 2 ( E X ) 2 ≤ 1 + ∑ i = 0 m − 1     p i − m ( m − i m ) q r ⌊ i n m ⌋ [ r n ] ≤ 1 + ∑ i = 0 m − 1     p i − m ( m − i m ) q r ⌊ i n m ⌋ q n ( r − n )

Since [ r n ] ≥ q n ( r − n ) . Thus

E X 2 ( E X ) 2 ≤ 1 + ∑ i = 0 m − 1     p i − m q − r n + r ⌊ i n m ⌋ ( m − i m ) q n 2 . (4)

Now consider p i − m q − r n + r ⌊ i n m ⌋ . We have

q − r n + r ⌊ i n m ⌋ ≤ q − r ( n − i n m ) = ( q r n / m ) i − m .

Thus p i − m q − r n + r ⌊ i n m ⌋ ≤ ( p q r n m ) i − m . But lim r → ∞ p q r n m = ∞ , hence lim r → ∞ ( p q r n / m ) i − m = 0 for 0 ≤ i ≤ m − 1 . It follows from Equation (4) that lim r → ∞ sup E X 2 ( E X ) 2 ≤ 1 ; hence lim r → ∞ E X 2 ( E X ) 2 = 1 . Therefore, by Chebyshev’s Inequality, lim r → ∞ P ( X ≠ 0 ) = 1 . We conclude that q − r n m 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 ω r has an n-element independent set is q − r .

Corollary 2 If m is a fixed positive integer exceeding two, then a threshold

function for the property that ω r has an m-element circuit is q − r ( m − 1 ) m .

Corollary 3 If n is a fixed positive integer, then a threshold function for the property that ω r contains a submatroid isomorphic to P G ( n − 1 , q ) is

q − r n ( q − 1 ) q n − 1 .

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 U n , n . Corollary 2 requires one to verify that an m-element circuit is k-balanced; this is precisely the uniform matroid U m − 1 , m , while in Corollary 3, the projective geometry P G ( n − 1 , q ) 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] .

Al-Hawary, T. (2017) On Functions of K-Balanced Matroids. Open Journal of Discrete Mathema- tics, 7, 103-107. https://doi.org/10.4236/ojdm.2017.73011