A conjecture of Welsh revisited

Welsh conjectured that for any simple regular connected matroid M, if each cocircuit has at least 12(r(M)+1) elements, then there is a circuit of size r(M)+1. This conjecture was proven by Hochstattler and Jackson in 1997. In this paper, we give a shorter proof of this conjecture based solely on matroid-theoretical arguments. Let M be a simple, connected, regular matroid and let C@?C(M), where |C|@?min{r(M),2d-1}. We show that if |C^*|=d=2,@?C^*@?C^*(M) where C@?C^*=0@?, then there is a circuit D such that D@?C is a circuit where |D@?C||C|.