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Find the minimum k

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September 2, 2010
combinatoricsExtremal combinatoricsSet systemspartitionIMO ShortlistIMO Longlist

Problem Statement

Let n≥2n \geq 2 be an integer. Find the minimum kk for which there exists a partition of {1,2,...,k}\{1, 2, . . . , k\} into nn subsets X1,X2,⋯ ,XnX_1,X_2, \cdots , X_n such that the following condition holds: for any i,j,1≤i<j≤ni, j, 1 \leq i < j \leq n, there exist xi∈X1,xj∈X2x_i \in X_1, x_j \in X_2 such that ∣xi−xj∣=1.|x_i - x_j | = 1.
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