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 − 1 k | E ( G ) | edges.