  We prove that every 3-edge-connected graph $G$ has a 3-flow $\phi$ with the
property that $|\mathop{supp}(\phi)| \ge \frac{5}{6} |E(G)|$. The graph $K_4$
demonstrates that this $\frac{5}{6}$ ratio is best possible; there is an
infinite family where $\frac 56$ is tight.
