On bipartite restrictions of binary matroids

In a 1965 paper, Erdős remarked that a graph G has a bipartite subgraph that has at least half as many edges as G. The purpose of this note is to prove a matroid analogue of Erdős's original observation. It follows from this matroid result that every loopless binary matroid has a restriction that uses more than half of its elements and has no odd circuits; and, for 2≤k≤5, every bridgeless graph G has a subgraph that has a nowhere-zero k-flow and has more than k−1k|E(G)| edges.