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Rainbow arithmetic progressions and anti-Ramsey results. (English) Zbl 1128.11305

Summary: The van der Waerden theorem in Ramsey theory states that, for every \(k\) and \(t\) and sufficiently large \(N\), every \(k\)-colouring of \([N]\) contains a monochromatic arithmetic progression of length \(t\). Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of \([3n]\) contains a 3-term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of \(\mathbb{Z}_n\). Finally, we give a general perspective on other anti-Ramsey-type problems that can be considered.

MSC:

11B25 Arithmetic progressions
05D10 Ramsey theory
11P70 Inverse problems of additive number theory, including sumsets
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