PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 46, Number 1, October 1974
ANOTHER EXCHANGE PROPERTY FOR BASES
CURTIS GREENE 1
ABSTRACT.
Let
X and
Y be bases
of a combinatorial
geometry
of
rank n.
If A C X and B C Y, with \A | + |ß| > n + 1, then there exist
subsets
A CAandB
CB
such that (X - A Q) u BQ and ( Y - B Q) U
A, are both bases.
In this
note
we prove
the following
result,
originally
of a vector
space
conjectured
by G.-
C. Rota.
Theorem.
Let X and
Y be bases
of finite
dimension
n.
If A C X and BCY, with \A\ + \B\ >n + 1, then there exist subsets A C A
and BQ Ç 73 such that (X - A ) US and (Y - B ) U A Q are both bases.
It will be clear
from the proof that the result
of a combinatorial
proved
geometry.
by the author
We will
use
i ' ' * > y i>-
(circuit)
determined
y = Y - y, the
C.=
y.uA.uA'.
i
J i
i
linear
—
Proof.
suppose
"exchange"
property
notation:
by' Jy i. and X. We also write
closure
we let F = Í5 C ß U.C¿ac
Lemma
Y are bases
•••, xn\, Y = ¡y,,---,
yn\, A = \xk, • • •, xj,
B =
each z = 1, • • •, k, let C . be the minimal dependent
F°r
(X - A),' so that
to another
if X and
[l], but the proof is much more elementary.
the following
Let X = x
\y
It is related
holds
'
t
5
For each 5Cß,leta.=
—
'
i
of Y - y.
set
A i. = C i. D A,' A i. = C.i D
(Equivalently,
S
A
cy, where
y ei J '
a^ = Y - 5.)
Finally,
for all y. £ S\.
z
1. F zs nonempty.
Suppose
not.
Then,
relabelling
the y.'s
if necessary,
we may
that
^i^V
'42ÇaBVy1,
A3caßv
AA<aBvyiV—
yi Vy2,-,
V yA.t-
Received by the editors July 15, 1973.
4A/S (MOS) subject classifications
(1970). Primary 05B35; Secondary
15A03,
15A15.
Key words
and phrases.
Combinatorial
geometry,
matroid.
1 Research supported in part by ONR N00014-67-A-0204-0063.
Copyright © 1974, American Mathematical Society
155
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156
CURTIS GREENE
(Choose
y7 such
that A
< a
; remove
it, and continue,
etc.)
But then,
if
/3 = aB V x V ••• V xk_x, we have
This
implies
that
ß > A
and
A',
and hence y ,
|8 > A
and
A',
and hence y ,
/3 > A
and
A',
and hence y .
ß > y , y ,•••, y
and a
, which
is impossible,
since
K/S)<8-1.
Lemma
2. Any minimal
member
Proof.
We may suppose
that
follows
such
trivially.
that
|5| > 1, since
If y. £ S, then,
A . C a„_
= a_
S of F is exchangeable.
since
V y-
otherwise
S - y . Í F, there
If we write
; = f(i),
the conclusion
exists
then
y. £S - y.
it is clear
that
2
/: S —»S is a bijection.
(a_
V y')=
an element
clear
(Since f(i) = /(/') = / => A . < (a^, V y •) A
ac> contradicting
z . £ A .... — a.,
2
/(z)
S
that S may be replaced
the fact
Since
exchanged
y.'s
successively
to X and removing
and so the same
1
the z ,'s does
that 5 e F.)
^
For each
y.£S,
V V ■ = clc V ^ . for each
' 2
ç.
A .- a
for the y .'s.
elements
S
by {z .\
A . for any / 4=f(i), since the sets
7
ar
s
y
Moreover,
are disjoint.
(That
Hence,
is, successively
not alter
are exchangeable.)
¡
z. does
^ z
completes
'
not appear
in
the z ,'s may be
2
adding
any of the remaining
This
choose
y . e 51, it is
the
circuits,
the proof.
REFERENCE
1. C. Greene,
A multiple
exchange
property
for bases,
Proc.
Amer.
Math.
Soc.
39 (1973), 45-50.
DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY,
CAMBRIDGE, MASSACHUSETTS02 139
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