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union of mutually disjoint 3-element subsets >= n-5

Source: Czech-Polish-Slovak Match 2014 day 2 P3

October 7, 2017
floor functioncombinatoricsSubsetsgeometrygeometric transformation

Problem Statement

Let n≥6n \ge 6 be an integer and FF be the system of the 33-element subsets of the set {1,2,...,n}\{1, 2,...,n \} satisfying the following condition: for every 1≤i<j≤n1 \le i < j \le n there is at least ⌊13n⌋−1 \lfloor \frac{1}{3} n \rfloor -1 subsets A∈FA\in F such that i,j∈Ai, j \in A. Prove that for some integer m≥1m \ge 1 exist the mutually disjoint subsets A1,A2,...,Am∈FA_1, A_2 , ... , A_m \in F also, that ∣A1∪A2∪...∪Am∣≥n−5|A_1\cup A_2 \cup ... \cup A_m |\ge n-5
(Poland)
PS. just in case my translation does not make sense, I leave the original in Slovak, in case someone understands something else
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