Given $\epsilon\in(0,1)$, suppose we have collection $\mathscr{C}(\epsilon)$ of multilinear polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ is in range $[-\epsilon,\epsilon]$ on $S_0$ while being in range $[1-\epsilon,1+\epsilon]$ on $S_1$ where $S_0\cap S_1=\emptyset$ with $S_0\cup S_1=\{0,1\}^n$.
Is this collection compact?