On pancyclic representable matroids (Q2581425)

A simple graph \(G\) with vertex set \(V(G)\) is pancyclic if it contains cycles of all lengths \(l\), for \(3 \leq l \leq | V(G| \). \textit{J. A. Bondy} [J. Comb. Theory, Ser. B 11, 80--84 (1971; Zbl 0183.52301)] proved that an \(n\)-vertex simple Hamiltonian graph with at least \(n^2/4\) edges is pancyclic unless it is isomorphic to \(K_{n/2,n/2}\). The authors address the question of an analog to Bondy's theorem for binary or GF\((q)\)-representable matroids; hence, this paper considers finding circuits of every size in GF\((q)\)-representable matroids with large numbers of elements. A consequence of the main result is that a rank-\(r\) simple binary matroid with at least \(2^{r-1}\) elements either has circuits of all sizes or is isomorphic to AG\((r-1,2)\).