1. Prove that if there exists some h ∈Hnk that has zero error over S(G) then G is k-colorable. Hint: Let h = !k

j=1 h j be an ERM classifier in Hnk over S. Define a coloring of V by setting f (vi ) to be the minimal j such that hj (ei ) = −1. Use the fact that halfspaces are convex sets to show that it cannot be true that two vertices that are connected by an edge have the same color.

2. Prove that if G is k-colorable then there exists some h ∈ Hn k that has zero error over S(G). Hint: Given a coloring f of the vertices of G, we should come up with k hyperplanes, h1 . . .hk whose intersection is a perfect classifier for S(G). Let b = 0.6 for all of these hyperplanes and, for t ≤ k let the i ’th weight of the t’th hyperplane, wt,i, be −1 if f (vi ) = t and 0 otherwise.