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2 colors for numbers 1,2,3,..., 2^n

Source: Czech and Slovak Olympiad 1988, National Round, Problem 4

September 13, 2024
combinatoricsColoringnumber theoryCombinatorial Number Theory

Problem Statement

Prove that each of the numbers 1,2,3,...,2n1, 2, 3, ..., 2^n can be written in one of two colors (red and blue) such that no non-constant 2n2n-term arithmetic sequence chosen from these numbers is monochromatic .
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