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2-element subests of 1-n

Source: Polish MO Recond Round 1991 p5

September 9, 2024
combinatoricsSubsets

Problem Statement

P1,P2,…,Pn P_1, P_2, \ldots, P_n are different two-element subsets of {1,2,…,n} \{1,2,\ldots,n\} . The sets Pi P_i , Pj P_j for i≠j i\neq j have a common element if and only if the set {i,j} \{i,j\} is one of the sets P1,P2,…,Pn P_1, P_2, \ldots, P_n . Prove that each of the numbers 1,2,…,n 1,2,\ldots,n is a common element of exactly two sets from P1,P2,…,Pn P_1, P_2, \ldots, P_n .
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