In mathematics, a uniform matroid is a matroid in which the independent sets are exactly the sets containing at most r elements, for some fixed integer r. An alternative definition is that every permutation of the elements is a symmetry.
The uniform matroid
A subset of the elements is independent if and only if it contains at most
is another uniform matroid
A uniform matroid is self-dual if and only if
Restricting a uniform matroid
and contracting it by one element (as long as
[5] The uniform matroid
may be represented as the matroid of affinely independent subsets of
points in general position in
-dimensional Euclidean space, or as the matroid of linearly independent subsets of
vectors in general position in an
-dimensional real vector space.
Every uniform matroid may also be realized in projective spaces and vector spaces over all sufficiently large finite fields.
[6] However, the field must be large enough to include enough independent vectors.
can be realized only over finite fields of
or more elements (because otherwise the projective line over that field would have fewer than
For this reason, uniform matroids play an important role in Rota's conjecture concerning the forbidden minor characterization of the matroids that can be realized over finite fields.
[7] The problem of finding the minimum-weight basis of a weighted uniform matroid is well-studied in computer science as the selection problem.
It may be solved in linear time.
[8] Any algorithm that tests whether a given matroid is uniform, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.
is connected: it is not the direct sum of two smaller matroids.
[10] The direct sum of a family of uniform matroids (not necessarily all with the same parameters) is called a partition matroid.
Every uniform matroid is a paving matroid,[11] a transversal matroid[12] and a strict gammoid.
[6] Not every uniform matroid is graphic, and the uniform matroids provide the smallest example of a non-graphic matroid,
-edge dipole graph, and the dual uniform matroid
is the graphic matroid of its dual graph, the
is the graphic matroid of a graph with
Other than these examples, every uniform matroid
-point line provides an example of a Sylvester matroid, a matroid in which every line contains three or more points.