Uniform matroid

In mathematics, a uniform matroid is a matroid in which every permutation of the elements is a symmetry. The uniform matroid U n r {displaystyle U{}_{n}^{r}} is defined over a set of n {displaystyle n} elements. A subset of the elements is independent if and only if it contains at most r {displaystyle r} elements. A subset is a basis if it has exactly r {displaystyle r} elements, and it is a circuit if it has exactly r + 1 {displaystyle r+1} elements. The rank of a subset S {displaystyle S} is min ( | S | , r ) {displaystyle min(|S|,r)} and the rank of the matroid is r {displaystyle r} . A matroid of rank r {displaystyle r} is uniform if and only if all of its circuits have exactly r + 1 {displaystyle r+1} elements. The matroid U n 2 {displaystyle U{}_{n}^{2}} is called the n {displaystyle n} -point line. The dual matroid of the uniform matroid U n r {displaystyle U{}_{n}^{r}} is another uniform matroid U n n − r {displaystyle U{}_{n}^{n-r}} . A uniform matroid is self-dual if and only if r = n / 2 {displaystyle r=n/2} . Every minor of a uniform matroid is uniform. Restricting a uniform matroid U n r {displaystyle U{}_{n}^{r}} by one element (as long as r < n {displaystyle r<n} ) produces the matroid U n − 1 r {displaystyle U{}_{n-1}^{r}} and contracting it by one element (as long as r > 0 {displaystyle r>0} ) produces the matroid U n − 1 r − 1 {displaystyle U{}_{n-1}^{r-1}} . The uniform matroid U n r {displaystyle U{}_{n}^{r}} may be represented as the matroid of affinely independent subsets of n {displaystyle n} points in general position in r {displaystyle r} -dimensional Euclidean space, or as the matroid of linearly independent subsets of n {displaystyle n} vectors in general position in an ( r + 1 ) {displaystyle (r+1)} -dimensional real vector space. Every uniform matroid may also be realized in projective spaces and vector spaces over all sufficiently large finite fields. However, the field must be large enough to include enough independent vectors. For instance, the n {displaystyle n} -point line U n 2 {displaystyle U{}_{n}^{2}} can be realized only over finite fields of n − 1 {displaystyle n-1} or more elements (because otherwise the projective line over that field would have fewer than n {displaystyle n} points): U 4 2 {displaystyle U{}_{4}^{2}} is not a binary matroid, U 5 2 {displaystyle U{}_{5}^{2}} 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.