ASymptotic properties of random subsets of
projective spaces
•
Douglas G. Kelly * and James G. Oxley+
Departments of Statistics and Mathematics
University of North Carolina
Chapel Hill, NC 27514
Abstract
A random graph on
on
n
vertices.
n
vertices is a random subgraph oD the complete graph
By analogy with this, the present paper studies the
asymptotic properties of a random submatroid
PG(r-l,q).
The main result concerns
geometry occurring as a submatroid of
one, for sufficiently large
r,
K
r
W
r
of the projective geometry
K, the rank of the largest projective
r
W.
r
We show that with probability
takes one of at most two values depending
~
on
r.
This theorem is analogous to a result of Bollobas and Erdos on the
clique number of a random graph.
However, whereas from the matroid theorem
one can essentially determine the critical exponent of
w ' the graph theorem
r
gives only a lower bound on the chromatic number of a random graph.
Keywords:
random graph, random submatroid, critical exponent.
*Partially supported by ONR Grant No. N00014-76-C-0550.
+partially supported by a Fulbright Postdoctoral Fellowship. Current address:
Mathematics Department, lAS, Australian National University, Canberra.
1
A.
Introduction
A random submatroid of a matroid
M is obtained from
a set of independent trials, one for each element of
M by performing
!1, at which the element
is deleted with probability l-p and retained with probability
p.
In the
study of random graphs such a process is used starting with the complete graph
on
n
vertices; every simple graph on
the experiment.
n
vertices is a possible outcome of
There are no matroids which are analogous to complete graphs
in this sense and so we choose to begin with projective geometries, the random
submatroids of which can be thought of as random simple matroids representable
over a given finite field.
A more complicated model for generating random
matroids was proposed by Knuth [7] and implemented by Cravetz [4].
However,
this approach does not seem easily amenable to probabilistic analysis.
The theorems of this paper may be informally summarized as follows.
•
a prime power
q
and for
jective geometry of rank
r
= 1,2, ... ,
rover
whether we assume the matroids
be the random submatroids of
GF(q).
M
M '}!2""
l
independent sets of size
Section D).
p
We shall assume that
we derive the expected values in
denote
M
r
PG(r-l,q), the pro-
Our analysis is unaffected by
to be nested or disjoint.
r
pendent trials as described above,
of an element.
let
r
Let
obtained by performing sets of indebeing the fixed probability of retention
0 < P < 1.
w
Fix
For any sequence
k ,k , ... ,
l 2
of the numbers of circuits of size
k , flats of rank
r
k ,
r
k , and bases (Proposition 1 and
r
In the cases of the numbers of circuits and independent sets, we
show that with probability one these random variables are asymptotic to their
expected values (Theorem 3).
A consequence of this is Theorem 4 which implies
that with probability one there is
circuit of size
such that each
r + 1, and therefore has rank
last section we consider the random variable
r
w
r
for
and is connected.
has a
In the
L, the rank of the largest
r
2
subspace of
M
r
all of whose elements are deleted.
ability one, for all sufficiently large
r,
L
We show that with prob-
takes its value in a set
r
which contains either a single integer or a pair of consecutive integers.
•
the critical exponent
made about
c
r
c
r
wr
of
(Theorem 7).
is just
V
r
Since
r - L
a similar statement can be
r'
Curiously, the asymptotic value of c
r
is
r - log r + o(log r), and only lower-order terms in the asymptotic expansion
q
q
involve the value of
p.
The proofs in the last section parallel those of Grimmett and
~1cDiarmid
[6], Matula [8,9],and Bo110bas and Erdos [3] for analogous results on random
graphs.
A summary of many of these graph-theoretic results appears in Bo1lobas's
book [2].
It should be noted that in the area of random graphs the terminology
used in limiting results is not uniform.
In particular, if
sequence of events, some authors use the term "A
merely that
1 - peA ) approaches zero as
n
n
n
A1 ,A Z' ...
is a
occurs almost surely" to mean
approaches infinity.
We have
stated our theorems using the term "with probability one"; such theorems are
true strong laws in the probabilistic sense.
In general we shall follow Welsh [11] for all matroid terminology
is otherwise unexplained.
be useful.
Some notation and a few simple inequalities will
Remembering that
h
r
1Mr I
which
q
is fixed, we define
qr _ 1
=
q - 1
1) (qr-l _ l) ... (qr-k+l
o
if
k < 0
or
1) ,
k=l,Z, ... ,r
k > r
Evidently,
We will be concerned with the asymptotic growth of the above quantities as
increases, for various choices of
k
depending on
r.
The obvious
r
3
inequalities
q
j-l
q
~
j
- 1 < q
j
for
1,2, ...
j
and
m
S-":..!.
n
q -1
~q
m-n
m> n
if
(1)
imply that
q
k(r-k)
<
[r]
k
~ q
k(r-k+l)
(2)
We also have
k
q
kr-(2)
k
~ [r]k ~ q
kr-(Z)-k
(3)
To sharpen these bounds we notice that
where
Obviously
Hr,k
~
which approaches
1.
1
For lower bounds we observe first that
as
r
tends to infinity if
Regardless of the growth of
-r+k
approaches
o.
k, we can obtain a lower bound by using the
inequality TT(l - a ) > 1 - Lan
n -
n
Hr k~
,
n=l
Even though the simpler bound
o<
(for
00
product is never zero.
kq
(1 - q
-n
a
< 1) :
n00
) > 1 -
I
n=l
~
q - 1
is zero for
-n =~
q
q - 1
q = 2, the infinite
4
Combining the above for later reference:
(1 _ q-n) > q-2 q
- q-1
k
kr-( )
2
(4)
and
r
as
-+
00
if
kq
-r+k
-+ 0
(5)
We will use two standard theorems from probability:
Chebyshev's Inequality.
variance
VX
If
and expected value
X is a random variable with finite
EX, then for any
The First Bore1-Cantel1i Lemma.
> 0,
£
{Al'~""}
If
is a sequence of
00
events and
exists
nO
L P(A) is a convergent series, then with probability 1 there
n=l
n
such that none of the A with n 2: nO occurs. (That is,
n
P(U
n A~)
N=l n=N
00
00
1.)
As easy consequence of these theorems we have the following lemmas, which
we will use repeatedly.
Lemma A.
00
~
VX
n
n=l (EX)
n
Let
2
(Xl 'X 2 '···)
is a convergent series.
If
lim inf EX
n
n-+oo
X
n
is positive for all
Proof:
For
X
such that
be a sequence of random variables, and suppose
I E~
Then
X
lim ---!l = 1
n-+oo EXn
with probability 1.
is positive, then with probability 1 there is
n
such that
2: nO'
k = 1,2, ... , let
1
1:.k
- 1 1<
nO
for
~
be the event that there exists
By Chebyshev's Inequality, the
n
Bore1-Cantel1i Lemma, and the hypothesis,
P~
1.
Therefore
5
X
n
00
1
= P(r-, ~) = P(lim EX = 1).
k=l
n+oo
that i f
EXn
Lemma B.
If
is positive, then
is finite, then
VX
To prove the second assertion we note
n
P(X
< 0) <
n-
P(X
= 0)
-
p(lxn VX
< ------
- (EX)2
EX
n
I -2
> !IEX /)
n
. 0
2
EX
= -----(EX)2
1.
o
Finally, the definition of expectation obviously implies
Lemma C.
If
P(X ; 0) < EX.
X is a nonnegative integer-valued random variable, then
0
6
B.
Some quantities associated with projective spaces
It is well-known (see, for example, [5]) that
of rank-k subspaces of
M.
numerical invariants of
M
[~]
equals the number
In this section we shall determine the other
r
that will be used in the remainder of the paper.
r
We shall need the following
Lemma D.
If
elements
x
Proof:
space
B
of
is a basis of
M, then there are precisely
M
B u x
r
V(r,q).
if the vector
is a circuit.
M
r
x
coordinates can be
It is clear that
B u x
has no zero coordinates.
x
is
chosen in
is a circuit of
Hence if
B u x
B
is the natural
M
r
q -1
if and only
is a circuit,
I, while each of the remaining
r - 1
ways from among the non-zero elements of
0
We now count the members of
1
r,k
and
collections of k-element independent sets and
A k-element independent set
rank
as the submatroid of the vector
Then, by sYmmetry, we may assume that
the first coordinate of
GF(q).
r-l
consisting of those non-zero vectors whose first non-zero
coordinate is one.
basis of
such that
We view the projective space
V(r,q)
(q-l)
r
k, namely its closure, I.
of
I
Mr
k-element circuits of
M.
r
lies in precisely one flat of
Therefore
}~
But
l ,k
k
for
example, [II, Exercise 16.1.4]) that
is the set of bases of
Cr, k which are respectively the
and it is not difficult to show (see,
(6)
7
I t follows that
11 r,k I =
Now, in
k > 3.
and
~-l'
1
(q-l)
kkT q
.
k
(2)
(7)
[r]k
ICr,k l , we note first that
To determine
Thus suppose
1
ICr,k l
=a
for
k < 3.
Then
consider the set of ordered pairs
C is a circuit containing
B.
(B,C)
where
B is a basis
By counting the number of such
pairs in two different ways, first over circuits and then over bases, we get,
using Lemma D, that
(q-l)
k-2,
I
Ik-l,k-l.
Thus, by (6),
and so
k-l
1
=
ICr,k'
Now suppose that
0
(2)
q-l k! q
equals
[r]k_l
for
Cr,k or 1r, k'
Then for
0, the number of members of
0
and
in exactly
elements does not depend on the choice of
this number
when
o = Cr,k
and
when
v=
(8)
k > 3.
{O,1,2, ... ,k}
i
D in
1
i
in
which meet
D.
1
.
r,k
D
We shall call
These numbers
arise in second moment calculations in the next section and the following
result bounds them above.
Lemma E.
ex.
1
~
{
(k~i)!
if
a
1,
if
i
< i < k - 1
k,
8
and
1
Proof:
Clearly
a
k
= 1.
fixed k-e1ement circuit of
k i [r-i]k_i
(q-l) -
for all
We now assume that
i < k
~1.
r
It is clear that
a.].
and
(k-i)-e1ement set
a k-e1ement circuit meeting
X in
Moreover, if
where
N
1
(ii)
Now
for all
j
in
(k-i)-tup1es
to
{Pl,P Z"" ,P j }; and
Y u
{P1,PZ"",Pk-i-1} u {P k- i }
r
- h.)(h
].
(P1,PZ, ... ,Pk-i)
is a circuit.
r
Therefore
and thus
k-1
i
.
1
(k) ( Z )-(Z)( _1)].-1[
']
(k-i)! i q
q
r - ]. k-i-1
The last expression is the stated bound on
a •
i
X be a
X and
Yu Z
in
Z, then
On using Lemma D, we obtain that
(h
Y so that
{l,Z, ... ,k-i-l}, the element
Y u
Y of
Y can be chosen
is the number of choices for
NZ is the number of
(i)
Y.
Z
let
is equal to the
product of the number of ways to choose an i-element subset
the number of ways to add a
in {O,l,Z, ... ,k}.
i
Pj
such that
is not in
is
ways.
9
B
To obtain the bound on
we use an argument similar to the above
i
to get
and rewriting the right-hand side of this, we obtain the required bound. 0
The last result of this section specifies one further quantity which
will be needed in a second moment calculation.
of rank-k subspaces of
of rank
M
r
Define
to be the number
which meet a fixed rank-k subspace in a subspace
Then i.t is not difficult to show (see, for example, [1, p. 225])
L
that
(9)
c.
Existence of circuits and independent sets
Let
{k}
r
be an arbitrary sequence of positive integers which we
will regard as fixed.
I
For simplicity we denote the families
and
C
r,k
r
by C and I
We also define the random variables C
and I
r
r
r
r to be
r
the numbers of kr-element circuits and k -element independent sets in w
r
r
r,k
Notice that a
if and only if
the elements of
in
J
is a circuit (resp. independent set) in W
r
is a circuit (resp. independent set) in
J
is deleted.
J
and none of
M
So if we define, for each
r
k -set
r
J
M ,
r
X =
J
..
k -set
r
f
if none of the elements of
1,
0, otherwise,
J
is
deleted,
(10)
10
then
L
C
r
Moreover,
EX
JEC
P(X
J
X
J
and
I
L
r
JEI
r
p IJI .
1)
J
k
k
r
r
Therefore we have by (8) and (7)
Proposition 1.
EC
X
J
Lk!
1
P rlCrl = q-l
q
k -1
( r )
2
r
> 3
[r]k -1 provided k r r
(11)
and
k
EI
r
p r II
r
Y
1
I
k
(q-l)
k
(2r)
kr
r
q
[r]k
(12)
r
0
The central result of this section is
00
Proposition 2.
VC
L
00
r
and
r=2 (EC ) 2
r
r
For every choice of the sequence
if
3<k
<r+l
if
o<
< r
for all
r-
are convergent series.
r=l (EI ) 2
r
As a corollary of Proposition 2 we get, using Lemma A,
The proof is given below.
Theorem 3.
L
VI
{k },
r
C
r, the~with probability one,
I
k
r-
for all
lim _r_ = 1;
r-to:> EC r
r, then, with probability one,lim E~ = 1.
r-to:>
r
Proposition 1 together with (4) and (5) provide asymptotic expressions for
EC
r
and
EI, which are almost-sure asymptotic values of
r
Since
EC
r
and
EI
C
r
and
I .
r
are bounded away from zero, we also have from
r
Lemma A:
Theorem 4.
•
if
For every choice of the sequence
3 < k
r
O
r
< r + 1
such that
for all
w
r
has a
{k },
r
r, then with probability 1 there exists
k -circuit for all
r
r':' r O ;
11
if
1 < k
< r
r-
for all
wr
such that
r, then with probability
has a
In particular, if we choose
circuit of size
r + 1
there exists
r > r .
- 0
k -independent set for all
r
k
r
=r +1
for circuits we see that with
w
such that for all
probability 1 there exists
•
1
has a
r
and thus is connected and has rank
r•
Proof of Proposition 2.
2
EC r
I"'
P(X X
L
J1 J
Z
JZEC r
L
JIEe r
L
1)
JlEC r
L
p
ZkrlJlnJzl
J EC
2 r
(for any fixed
p
where
i
a,
1
is the number of
-i
A.,
1
k -circuits intersecting a fixed k -circuit in
r
r
points.
Therefore,by Lemma E and (8),
(EC ) Z
r
k -1
r
_(i)
k!
2
r
= 1 + L P
(k -i)! (:r) q
r
i=l
_ (i)
k r -1
k!
-i
2
r
:S 1 + L P
q
(k -i)!
i=l
r
-i
(q-l)
[r]i
q
i
9
+
ir-(i)-i
2
k -1
VC
<
L
(EC )2 - i=l
r
k !
r
~
p
k -1
k
qk
r
+
t
i
r
p
r
_r_
pq
r-l ]
q-
er~l)
r
Therefore
•
r
k !
+ ....L.
k
P
(»
where the last step follows by (3).
i
q
(9-1 )
[r]k -1
r
9
(k -l)(r-l)
r
12
where
p
k!
k
(k -i)!
i
-i ~~r~ (r)
r
q
-i(r-2)
Now
(k _i)2
r
= -p1 i+1
•
t.+ /t. < 1
1 1
1 -
and thus
VC
q
- (r-2)
2
r
<~,
pq
for sufficiently large
(EC )
k
__
r_
qk
r
2
< k
t
r 1
r.
+---!:
p
pq
r
r-1
So for sufficiently large
k -1
r
]
(r+1)
<
r-2
pq
3
q(r+1)
p
+
r-1
[--E±L]
pq
r,
2
.
· 1S
. t he r t h
Th 1S
term.1n a convergent ser1es.
Turning now to independent sets, we proceed almost exactly as for circuits.
Ir r Ip
where
Si
r
i
r
r
I
i=O
k -independent sets intersecting a fixed
is the number of
k -independent set in
k
2k
r
points.
Therefore,by Lemma E and (7),
k
k
r
L
i=O
k
1
+
I
p
k
-i
.
(r)_(1)
2
2
r
(k -i) I (i ) q
r
r
(q-1)
1 k -1 [r-i]k -1
(q-l) r
r
i
[r]i
i=l
This differs only slightly from the upper bound obtained on
in the argument
(EC ) 2
above.
that
.
"
D.
A straightforward modification of that argument shows
VIr
th
2 is the r
term in a convergent series. o
(EI
r
r
)
Expected numbers of bases and flats.
Again we consider as fixed a given sequence
{k}
r
of positive integers;
.~
\
13
and we define the families
rank
k)
r
bases and
in
B
and
r
of bases and k -flats (flats of
r
M, and the random variables
r
k -flats in
Band
r
F, the numbers of
r
w.
r
r
Notice that the results of the previous section imply the existence
with probability 1 of an
and therefore
r
O such that
w
r
has full rank
for all
r '
O
~
r
almost surely equals Irr,r l for large r. In this sec-
Br
tion we find the expected values of
B
r
and
F
in terms of the Tutte
r
polynomials (see [11, Chapter 15])of the underlying projective geometries
We do not obtain asymptotic results.
Mi'
The expected values are given in (16)
and (17).
Bases.
r
EB
I
r
E(B
i=O
I
r
rank(w )
i) ,
r
and
E(B
rank(w )
r
r
I
E(B
JE: M.
=
i)
I
rank(w )
i
r
r
and
w c J)P(w
r r
c
J
I
rank(w )
r
l.
is the family of rank-i subspaces of
("There
for any fixed rank-i subspace J
O
of Mr'
Now such a
M)
is isomorphic to
r
M. ,
l.
so an argument similar to that used for Proposition 1 shows that this last
quantity equals
p
i
times the number of i-independent sets in
E(B
•
To find
Welsh [10].
r
P(rank(w)
r
If
I
i
rank(llJ )
=
r
i)
i)
= E-
q
(~)
M ; that is,
i
[i].
l.
i!
(q_1)i
we use the following theorem of Oxley and
M is a matroid of rank
i
on
h
elements and w is
(13)
i)
14
a random submatroid of
= i) = pi(l
P(rank(w)
where
M, then
_ p)h-i T(M;l,(l-p)-l) ,
is the Tutte polynomial of
T(M;x,y)
P(rank(fJ.l )
Using this theorem:
P(all elements of
r
w
I Milp(Mr
- J
M - J
are deleted and
r
has full rank in
r
=
M.
(14)
J)
M
is de1eted)P(a random submatroid of
O
i
has
full rank)
Here
J
can be any fixed member of Mi
O
P(rank(w )
h -h
r i
[ r](l_p)
i
i)
r
= [ri ]
Combining (13) and (15)
r
EB r
= L
i=O
.
p
1
(l-p)
EF
r
i
P (1
-p)
h r -i
hi-i
T(
M1
; , (1 -p )
i
T(M.;l,(l-p»
-
~ (l-p) r
1.
q
r
•
= [k ] (l-p)
r
r
(15)
(q-1) i
i
=r
k -flats
r
h
-k
r
k
r
(16)
).
dominates this sum because
in
k
p rT(M
-1
1
w
r
r.
M
r
times
the probability that a given such flat has full rank in
EF
)
1
T (M. ; 1, (l-p)
for sufficiently large
equals the number of
r
-1
1
[ ]
( i)
2
r i
h -i
Notice . that the term corresponding to
Flats.
follows that
~ives
2i
almost surely has rank
It
w.
r
By (14),
-1
kr
;l,(l-p)
).
(17)
15
E.
Largest full subspace.
For
= 1,2, ... , Ie t
r
K
r
be the rank of the largest full subspace of
w . that is, the largest subspace of
M
r
r'
with no deleted elements.
Our main
result in this section is Theorem 6, which implies that with probability 1
there is
r > r
such that for all
two possible values.
K
r
has at most
Symmetry gives a similar result (Theorem 7) for the rank
of the largest subspace of
critical exponent of
the random variable
O
w •
r
M
r
with no retained elements, and hence for the
(It is merely for convenience of notation that our
results are proved for full rather than empty subspaces.)
For an arbitrary integer
of
k, let
F
r,k
be the family of rank-k subspaces
M . then
r'
Let
N
r,k
be the number of full rank-k subspaces of
As with circuits and
w •
r
independent sets,
Nr,k =
where
X
J
is defined by (10).
Therefore, for any
1)
EN r, k
Moreover,
K
r
< k
i f and only i f
J
in
[~]p
Fr, k'
h
k
N
r, k = 0 •
In this section "log" '''ill denote base-q logarithms and "l n "
logarithms.
•
We also let
so that
k
b > 1 and
b
-q +1
natural
16
E > 0, define
For any
d
° < E < 1,
Notice that i f
by 1.
d
r,O
r,E
= llO
r log r + Ej .
log b
g
It can also be checked that if
and
d
r,E
d
then either
and
r,O
d
are equal or they differ
r,E
E is a given positive number and j and k denote
,then for sufficiently large
r,
ENr,j > 1 > ENr,k+l.
00
Proposition 5.
For any
L
> 0,
E
00
P(K
r=l
r
> d
r,E
)
L
and
P(K
r=l
r
are
< d r, 0)
convergent series.
The proof is given below.
Theorem 6.
Suppose
that for every
°
<
r > r '
O
As a corollary we get from the Borel-Cantelli Lemma.
Then with probability 1 there exists
E < l.
K
r
has its value in the set
{d
r,O'
d
r,E
}
r
O
such
(which may be
a singleton or a pair).
This theorem translates immediately by sYmmetry to a result on the rank
L
r
of the largest subspace of
exponent
c
of
r
wr ' where
c
H
with no retained elements and on the critical
r
= r - L
r
r
d'
r,E
E >
r log r
log b'
°
let
+ EJ
1
(_1_) q-1
h'
Suppose
that for every
°<
r > r O'
E < l.
L
r
and
l-p
Then with probability 1 there exists
c
r
and {r - d'
r
respectively.
r,E' - d'r, O},
•
For
llOg
where
Theorem 7.
.
have their values in the sets
r
O
such
{d' ,d'
}
r,O r,E
17
We note two more consequences of the above before proving Proposition 5.
Firstly, the asymptotic expressions for
that are independent of
K
L
r
K , L , and
r
r
c
have high-order terms
r
p:
d
r
- log r + o(log r), and
r,O
c
- r - log r + o{log r) .
r
This is in contrast to the growth of the size of the largest clique in a random
graph as found in [6,9,2].
Secondly, with probability one,
greater than two and hence for sufficiently large
over fields containing
r,
K
r
is eventually
is representable only
W
r
GF{q).
Proof of Proposition 5.
r
2
P{K
We prove that
r
> d
r,E:
+ 1) ~ 0 as
r
~
(I8)
00
and
r
2
P{K
r
< d
0) ~
r,
a
as
r ~
(19)
00,
and the proposition follows.
To prove (18) we notice that for any
P{K
r
> k)
=
peN
r,k
k, by Lemma C,
~ 0) < EN
r,k
q
= [r]bk
and sO,by (2),
P{K
Now if
k = d
r,E
+ 1, then
r log r qE:
log b
..
and
•
So
k
> k) < qk(r-k+l)b- q +1
r-
< q
k
<
r log r 1+ E:
log b q
k
+1
18
= [
q £: - 1 log b
r
which tends to
0
as
r
Next we prove (19).
+
For any
Ed
r £:
'
k, by Lemma B,
< k)
r
r
q
Thus (18) is proved.
00.
P(K
q 1+£:] r-d r ,£: _-,,-b--:r:...2__
log r
P(N r, k
Now
Z
EN r, k
=
I
"L
J 1 EF r ,k
P(X
JZE:Fr,k
I
IFr,k'
p
J
1
X
J
Z
1)
Zh k -IJ1 nJ Z '
(for any fixed
JZEFr,k
'e
k
Zh
[~]p k
where
Yi
-h
I
y.
i=O 1.
P
i
is the number of rank-k subspaces intersecting a fixed rank-k subNow,because of (9),
space in a rank-i subspace.
-1 +
Z
EN r,k
k
(Ett~Z
< -1 +
I
i=O
T
i
where
- q
Now,by (1),
TO
~
(k-i)
Z
i
q-1
b,
i
O,l, .•. ,k
1; and (Z) implies that
i
<,
J 1 E Fr,k)
T. < b
1. -
q -1 qk-i(r-Zk+i)
(i
1, Z, ... , k)
19
Therefore
k
< k) <
P(K
r
L si
i=l
where
b
Now we show that if
k
d
r,
/ -1 k-i(r-2k+i)
q
0' then for sufficiently large
r
the function
first decreases and then increases and has exactly one critical point in the
interval
1 < x < k.
It will follow that
d
We use the fact that if
k
f(x)
(20)
d r, 0' then
r log r < qk < r log r
- log b
q log b We can rewrite
and sufficiently large r.
r,O
and
< r
r
as
x
k (q -1)log b - x(r-2k+x)
q q
and it suffices to show that the nonconstant part of the exponent,
g(x)
= qx log
has the properties claimed above for
g' (x)
•
b - x
f(x).
2
- (r - 2k)x ,
But
x
q In b - 2x - r + 2d r, 0
so
g' (1)
= qlnb
- 2 - r + 2d
r,
0
(21)
20
which is obviously negative for large
k
q 1nb
g' (k)
r.
r > r log r 1nb - r
- q log b
= r(log
which is positive for large
for 1 < x < k, and so
[l,k].
But
Thus
g(x)
r)lnq - r
first decreases and then increases
g'(x), being continuous, has an odd number of zeros in
g'(x)
has at most two zeros, since it is the difference
between the convex function
g'(x)
r.
Moreover,
x
q 1nb
has exactly one zero in
proved, and (20) follows.
2x + (r - 2k).
and the linear function
[1, k] ,
the assertion about
f(x)
So
is
We get
2
q
b k 3k-r + 1- kb qk q k +k-kr
2bq
2b
=-q
•
r
2
q
q
b k 3k-r
2 b 1
r log r
< r 2bq og( log b)
2bq q
q
1
r-3k
-2
o(r
But we can use (21) to show that each of these terms is
-+
0
):
as
r -+
00,
and
10g(r 2
2~
10g10g(r1~;gbr
kbqkqk2+k-kr) < 2 log r - log 2b +
+ k
2
) + r log r
+ k - kr
4
r(log r - k) + o«log r) )
4
r log r
2 r(log r - 10g( log b ) + 1) + o«log r) )
-+ -
00
as
r
-+
00.
o
21
References
,
1.
George E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics
and its Applications, Volume 2 (Addison-Wesley, Reading, Massachusetts,
1976).
2.
B~la Bollob~s, Graph Theory. An Introductory Course, Graduate Texts in
Mathemotics No. 63 (Springer-Verlag, New York, Heidelberg, Berlin, 1979).
3.
B. Bollobas and P. Erdos, Cliques in random graphs, Math. Proc. Camb.
Phil. Soc. 80 (1976), 419-427.
4.
Allan E. Cravetz, Essentials for t1atroid Erection (M.S. Thesis, Department
of ~~thematics, University of North Carolina, Chapel Hill, 1978).
5.
Jay Goldman and Gian-Carlo Rota, The number of subspaces of a vector
space, Recent Progress in Combinatorics, Editor: W. T. Tutte (Academic
Press, New York, London, 1969) pp. 75-83.
6.
G. R. Grimmett and C. J. H. McDiarmid, On colouring random graphs,
Math. Proc. Camb. Phil. Soc. II (1975), 313-324.
7.
Donald E. Knuth, Random matroids, Discrete Math.
8.
David W. Matula, On the complete subgraphs of a random graph, Proc.
Second Chapel Hill Conf. on Combinatorial Mathematics and its APPlications
(U.N.C. Press, Chapel Hill, 1970) pp. 356-369.
9.
, Graph-theoretic cluster analysis, Classification and
Clustering, Editor: J. Van Ryzin (Academic Press, New York, San Francisco,
London, 1977) pp. 95-129.
10.
J. G. Oxley and D. J. A. Welsh, The Tutte polynomial and percolation, Graph
Theory and Related Topics, Editors: J. A. Bondy and D.S.R. Murty (Academic
Press, New York, San Francisco, London, 1979) pp. 329-339.
11.
D. J. A. Welsh, Matroid Theory, London Math. Soc. Monographs No.8
(Academic Press, London, New York, San Francisco, 1976).
11
(1975), 341-358.
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random graph, random submatroid, critical exponent
20.
ABSTRACT (ContInue
on reverSo
side If
ne{'O'>Sllry
and identify hy block
flumfHH)
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 w
..
r
of the projective geometry PG(r-l, q).
The main result concerns K , the rank of the largest projective geometry
r
occurring as a submatroid of w
We show that with probability one, for
DD
,:~:~, 1473
,""'0' 0' , ,ov"" 0':0''''
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