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Permutation of 1,2,...,n with no arithmetic subsequence

Source: Austrian-Polish 2004, Problem 6

July 5, 2015
arithmetic sequencePermutations with restrictionscombinatorics

Problem Statement

For n=2mn=2^m (m is a positive integer) consider the set M(n)={1,2,...,n}M(n) = \{ 1,2,...,n\} of natural numbers. Prove that there exists an order a1,a2,...,ana_1, a_2, ..., a_n of the elements of M(n), so that for all 1≤i<j<k≤n1\leq i < j < k \leq n holds: aj−ai≠ak−aja_j - a_i \neq a_k - a_j.
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