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 ) , 2 d − 1 } . We show that if | C ∗ | ≥ d ≥ 2 , ∀ C ∗ ∈ C ∗ ( M ) where C ∩ C ∗ = 0̸ , then there is a circuit D such that D △ C is a circuit where | D △ C | > | C | .