Let H be a universal family of hash functions mapping [n] = {0, 1, . . . , n - 1} to [m] = {0, 1, . . . , m - 1}. Find the most general relation of m and n that guarantees the existence of a hash function in H that causes no collision when hashing [n] into [m].
(Hint: Consider that the probability of a random hash function from H causing some collision is strictly less than 1.)