1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

## Prove that if there exists some h ∈Hnk that has zero error over S(G) then G is k-colorable.

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 *(e*i *) =* *−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, *h*1 *. . .**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.