Welsh's problem on the number of bases of matroids (original) (raw)

Given (n,r,b)(n,r,b)(n,r,b), a triple of integers with 0<rleqn0<r\leq n0<rleqn and 1leqbleqbinomnr1\leq b\leq\binom{n}{r}1leqbleqbinomnr, does there exist a matroid of size nnn, rank rrr, with bbb bases? Such a matroid is called an (n,r,b)(n,r,b)(n,r,b)-matroid. This problem was raised by Dominic Welsh and is studied in this paper. We prove that when corank n−rleq3n-r\leq3n−rleq3, (n,r,b)(n,r,b)(n,r,b)-matroids exist for all such triples except (n,r,b)=(6,3,11)(n,r,b)=(6,3,11)(n,r,b)=(6,3,11), which is the counterexample found by Mayhew and Royle. Furthermore, we show that (n,r,b)(n,r,b)(n,r,b)-matroids exist for all rrr large relative to the corank n−rn-rn−r.