On the critical exponent of transversal matroids

Brylawski [2] has shown that loopless principal transversal matroids have critical exponent at most 2. Welsh [4] asks if a similar result holds for all transversal matroids. We answer in the affirmative by proving that all loopless transversal matroids have critical exponent at most 2. The terminology used here for matroids will in general follow Welsh 131. A rank r transversal matroid M is representable over GF(q) for some q. In this case the canonical simple matroid associated with M is isomorphic to a submatroid of PG(r 1, q) and if M is loopless this submatroid and M have the same critical exponent c(M; q). So without loss of generality we may consider only transversal submatroids of PG(r 1, q). A cyclic flat is one which is a union of circuits.