185 research outputs found
Odd circuits in dense binary matroids
We show that, for each real number and odd integer
there is an integer such that, if is a simple binary matroid with and with no -element circuit, then has critical
number at most . The result is an easy application of a regularity lemma for
finite abelian groups due to Green
Projective geometries in exponentially dense matroids. I
We show for each positive integer that, if \cM is a minor-closed class
of matroids not containing all rank- uniform matroids, then there exists
an integer such that either every rank- matroid in \cM can be covered
by at most sets of rank at most , or \cM contains the
\GF(q)-representable matroids for some prime power , and every rank-
matroid in \cM can be covered by at most sets of rank at most .
This determines the maximum density of the matroids in \cM up to a polynomial
factor
On minor-closed classes of matroids with exponential growth rate
Let \cM be a minor-closed class of matroids that does not contain
arbitrarily long lines. The growth rate function, h:\bN\rightarrow \bN of
\cM is given by h(n) = \max(|M|\, : \, M\in \cM, simple, rank-$n$). The
Growth Rate Theorem shows that there is an integer such that either:
, or , or there is a
prime-power such that ; this
separates classes into those of linear density, quadratic density, and base-
exponential density. For classes of base- exponential density that contain
no -point line, we prove that for all
sufficiently large . We also prove that, for classes of base- exponential
density that contain no -point line, there exists k\in\bN such
that for all
sufficiently large
The densest matroids in minor-closed classes with exponential growth rate
The for a nonempty minor-closed class of
matroids is the function whose value at an
integer is defined to be the maximum number of elements in a simple
matroid in of rank at most . Geelen, Kabell, Kung and Whittle
showed that, whenever is finite, the function
grows linearly, quadratically or exponentially in (with
base equal to a prime power ), up to a constant factor.
We prove that in the exponential case, there are nonnegative integers and
such that for all sufficiently large , and we characterise
which matroids attain the growth rate function for large . We also show that
if is specified in a certain `natural' way (by intersections of
classes of matroids representable over different finite fields and/or by
excluding a finite set of minors), then the constants and , as well as
the point that `sufficiently large' begins to apply to , can be determined
by a finite computation
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