Problem proposers are becoming professionals at hiding graph theory.

Let me try to make the problem statement more clear: In a film there are $n$ roles. For each $i$ ($1 \leq i \leq n$), the $i$th role has $a_i$ people trying out for that role, and no person is trying out for multiple roles. Every day, a tryout is held, in which exactly one competitor for each of the $n$ roles tries out. Suppose that $p$ is a prime number such that $p \geq a_1, a_2, \cdots, a_n, n.$ Show that in $p^k$ days of tryouts, we can ensure that any $k$ people trying out for different roles have acted for the film together ($k \leq n$). Solution: Clearly, we may assume that $a_1, a_2, \cdots, a_n, n$ are all $p$, since this would make our job "strictly harder." (In other words, if $a_i < p$ we can just take a subset of the tryouts for the case when $a_i = p$ and if $n < p$, then we just add an $a_{n+1}$ then ignore the extra roles in each tryout for $n+1$). In this case, we have $p$ roles and $p$ people vying for each role. Let's label the roles $r_0, r_1, \cdots, r_{p-1}.$ Observe that there are $p^k$ integer polynomials of the form $c_{k-1} x^{k-1} + \cdots + c_1 x + c_0$ where the $c_i$'s are all residue classes (mod $p$). Let $P_1, P_2, \cdots, P_{p^k}$ be these $p^k$ integer polynomials. Then, for each $P_i$, on the $i$th day of casting, we will let the $P_i(j)$th person try out for the role $r_j$. To see that this covers all groups of $k$ people in different roles, simply use Lagrange Interpolation, keeping in mind the fact that $k \leq p$.