 Growth bound for signs of m-term linear combinations

For $f={\sum }_{j=1}^m{\alpha }_j{g}_j$ with $g_j$ drawn from a binary class $\mathcal{G}$, fixing $(g_1,\dots ,g_m)$ leaves an $m$-dimensional linear sign class. Consequentlywhen $\mathcal{G}$ is finite, and in general