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2020 PUMaC Individual Finals A3

Source:

January 1, 2022
combinatorics

Problem Statement

Let nn be a positive integer, and let FF be a family of subsets of {1,2,...,2n}\{1, 2, ... , 2^n\} such that for any non-empty A∈F A\in F there exists B∈FB \in F so that ∣A∣=∣B∣+1|A| = |B| + 1 and B⊂AB \subset A. Suppose that F contains all (2n−1)(2^n - 1)-element subsets of {1,2,...,2n}\{1, 2, ... , 2^n\} Determine the minimal possible value of ∣F∣|F|.
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