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Partition of family of sets

Source: Iranian National Olympiad (3rd Round) 2004

January 9, 2009
combinatorics proposedcombinatorics

Problem Statement

F \mathcal F is a family of 3-subsets of set X X. Every two distinct elements of X X are exactly in k k elements of F \mathcal F. It is known that there is a partition of F \mathcal F to sets X1,X2 X_1,X_2 such that each element of F \mathcal F has non-empty intersection with both X1,X2 X_1,X_2. Prove that ∣X∣≤4 |X|\leq4.
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