At Least |s|

Let S be a finite set. Let k be a positive integer. Let A be a subset of Sk satisfying |A| = |S|k−1. Let B = Sk \ A. For every v ∈ Sk and every i ∈ {1, 2, . . . , k}, we denote by vi the i-th component of the k-tuple v (remember that v is an element of Sk, that is, a k-tuple of elements of S). Then, every v ∈ Sk satisfies v = (v1, v2, . . . , vk). Let F be the set of all pairs (a, b) ∈ A × B for which there exists an i ∈ {1, 2, . . . , k} satisfying ( aj = bj for all j 6= i ) 1. (Speaking less formally, let F be the set of all pairs (a, b) ∈ A× B for which the k-tuples a and b differ in at most one position.) Prove that |F| ≥ |S|k−1 · (|S| − 1).