Maximum-sized matroids with no minors isomorphic to \(U_{2,5}\), \(F_7\), \(F_7^-\), or \(P_7\) (Q2777898)

\textit{I. Heller} [Pac. J. Math. 7, 1351-1364 (1957; Zbl 0079.01903)] proved that a rank-\(n\) simple binary matroid \(M\) with no minor isomorphic to the Fano matroid or its dual has at most \(\binom{n+1}{2}\) elements with the bound being attained if and only if \(M\) is isomorphic to \(M(K_{n+1})\). Several authors have proved analogues and extensions of Heller's theorem. This paper proves a far-reaching extension of the theorem by determining the maximum number of elements in a rank-\(n\) simple matroid with no minor isomorphic to the 5-point line, the Fano matroid, the non-Fano matroid, or \(P_7\), the matroid that is obtained by deleting two points from the ternary affine plane. When \(n\geq 4\), this maximum is the same as that in Heller's theorem. Moreover, when \(n\geq 5\), no new extremal matroids arise. Much of the paper is devoted to determining the extremal matroids for \(n\leq 4\).