Large Circuits in Binary Matroids of Large Cogirth, I

LetF7denote the Fano matroid andebe a fixed element ofF7. LetP(F7, e) be the family of matroids obtained by taking the parallel connection of one or more copies ofF7aboute. LetMbe a simple binary matroid such that every cocircuit ofMhas size at leastd⩾3. We show that ifMdoes not have anF7-minor,M≠F*7, andd⩾(r(M)+1)/2 thenMhas a circuit of sizer(M)+1. We also show that ifMis connected,e∈E(M),Mdoes not have both anF7-minor and anF*7-minor, andM∉P(F7, e), thenMhas a circuit that containseand has size at leastd+1.