Unavoidable Sets of a Certain Size in a Finite Set

In this paper, we introduce the U-function on N, the set of natural numbers. U(k, n), for positive integers k, n with k ≤ n and n ≥ 3, counts the number of unavoidable sets of size k which are subsets of {1, 2, ..., n}, where the notion of avoidability is defined below. We use some straightforward observations as well as the results of Shan, Zhu, Dumitriu, and Develin to observe that U(1, n) = U(2, n) = 0 for all n, and to give a recursive formula for U(3, n). Further, we provide nontrivial lower bounds for U(k, n) for k ≥ 4. We conclude by giving conditions under which r-step sequences, defined in the text, are avoidable.