A conjecture of Welsh revisited

Welsh conjectured that for any simple regular connected matroid M, if each cocircuit has at least 1 2 (r(M) + 1) elements, then there is a circuit of size r(M) + 1. This conjecture was proven by Hochstättler 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∗ = Ø, then there is a circuit D such that D4C is a circuit where |D4C|> |C|.