I'm reading Oxley's book Matroid Theory, and I read something that is not trivial for me... The book says that every uniform matroid is also transversal, but I don't understand why! I know that a matroid is uniform if and only if all of its circuits have exactly $r+1$ elements, where $r$ is the rank of the matroid, but I can't associate this to the definition of transversal matroid. Can you please help me?
2025-01-13 11:42:35.1736768555
Uniform and Transversal Matroid
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A transversal matroid can be modeled as a bipartite graph, with elements on one side and buckets equal in number to the rank on the other side. A set is independent if and only if there is a matching, and a set of elements is a circuit if it is dependent but discarding any element will make it a independent.
Consider the transversal matroid below with 5 elements and rank 3, represented by a complete bipartite graph. Every set of size 3 or lower is independent, and every set of size 4 is a circuit. Therefore this is $U_{3,5}$.