Intersecting set systems and graphic matroids

Two simple proofs are given to an earlier partial result about an extremal set theoretic conjecture of Chung, Frankl, Graham, Shearer and Faudree, Schelp, Sós, respectively. The statement is slightly strengthened within a matroid theoretic framework. The first proof relies on results from matroid theory, while the second is based on an explicit construction providing an elementary proof.