# Uniform matroid

In mathematics, a **uniform matroid** is a matroid in which every permutation of the elements is a symmetry.

## Definition

The uniform matroid is defined over a set of elements. A subset of the elements is independent if and only if it contains at most elements. A subset is a basis if it has exactly elements, and it is a circuit if it has exactly elements. The rank of a subset is and the rank of the matroid is .^{[1]}^{[2]}

A matroid of rank is uniform if and only if all of its circuits have exactly elements.^{[3]}

The matroid is called the **-point line**.

## Duality and minors

The dual matroid of the uniform matroid is another uniform matroid . A uniform matroid is self-dual if and only if .^{[4]}

Every minor of a uniform matroid is uniform. Restricting a uniform matroid by one element (as long as ) produces the matroid
and contracting it by one element (as long as ) produces the matroid .^{[5]}

## Realization

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. For instance, the -point line can be realized only over finite fields of or more elements (because otherwise the projective line over that field would have fewer than points): is not a binary matroid, is not a ternary matroid, etc. 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]}

## Algorithms

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.^{[9]}

## Related matroids

Unless , a uniform matroid 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, . The uniform matroid is the graphic matroid of an -edge dipole graph, and the dual uniform matroid is the graphic matroid of its dual graph, the -edge cycle graph. is the graphic matroid of a graph with self-loops, and is the graphic matroid of an -edge forest. Other than these examples, every uniform matroid with contains as a minor and therefore is not graphic.^{[13]}

The -point line provides an example of a Sylvester matroid, a matroid in which every line contains three or more points.^{[14]}

## See also

## References

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^{6.0}^{6.1}Template:Harvtxt, p. 100. - ↑ Template:Harvtxt, pp. 202–206.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ Template:Harvtxt, p. 126.
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- ↑ Template:Harvtxt, pp. 48–49.
- ↑ Template:Harvtxt, p. 30.
- ↑ Template:Harvtxt, p. 297.