For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds: For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, all of the same size, let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which \[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\] a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$. b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.