Turán's triangle theorem and binary matroids (Q1117942)

A special case of a theorem of Turán is that a graph on v vertices, with no loops, parallel edges, or triangles, has no more than \(\lfloor v/2\rfloor \lceil v/2\rceil\) edges. This bound is achieved by a graph G if and only if \(G\cong K_{\lfloor v/2\rfloor,\lceil v/2\rceil}\). We generalize this result to binary maroids. In particular, we show that if \(M=(E,{\mathcal C})\) is a rank r, binary maroid with no minor isomorphic to the dual of the Fano matroid, and if \(| C| \geq 4\) \(\forall C\in {\mathcal C}\), then \(| E| \leq \lfloor (r+1)/2)\rfloor \lceil (r+1)/2\rceil\), unless \(M\cong R_{10}\). Moreover, if the bound is attained then M is isomorphic to the circuit matroid of \(K_{\lfloor (r+1)/2\rfloor,\lceil (r+1)/2\rceil}\).