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KJMO maximum value of m subsets

Source: 1999 KJMO P8

June 30, 2024
set theoryKJMOcombinatorics

Problem Statement

For Sn={1,2,...,n}S_n=\{1, 2, ..., n\}, find the maximum value of mm that makes the following proposition true.
Proposition There exists mm different subsets of SS, say A1,A2,...,AmA_1, A_2, ..., A_m, such that for every i,j=1,2,...,mi, j=1, 2, ..., m, the set Ai∪AjA_i \cup A_j is not SS.
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