Abstract
Let F 7 denote the Fano matroid and M be a simple connected binary matroid such that every cocircuit of M has size at least d ⩾3. We show that if M does not have an F 7 -minor, M ≠ F * 7 , and d ∉{5, 6, 7, 8}, then M has a circuit of size at least min{ r ( M )+1, 2 d }. We conjecture that the latter result holds for all d ⩾3.
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