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Minimum number of values in the union of sets

Source: Simon Marais Mathematics Competition 2023 Paper A Problem 3

October 14, 2023
combinatorics

Problem Statement

For each positive integer nn, let f(n)f(n) denote the smallest possible value of ∣A1∪A2∪⋯∪An∣|A_1 \cup A_2 \cup \dots \cup A_n| where A1,A2,A3…AnA_1, A_2, A_3 \dots A_n are sets such that Ai⊈AjA_i \not\subseteq A_j and ∣Ai∣≠∣Aj∣|A_i| \neq |A_j| whenever i≠ji \neq j. Determine f(n)f(n) for each positive integer nn.
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