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arXiv · 2102.08995

Integer colorings with no rainbow 3-term arithmetic progression

Xihe Li·Hajo Broersma·Ligong Wang
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Abstract

In this paper, we study the rainbow Erd\H{o}s-Rothschild problem with respect to 3-term arithmetic progressions. We obtain the asymptotic number of $r$-colorings of $[n]$ without rainbow 3-term arithmetic progressions, and we show that the typical colorings with this property are 2-colorings. We also prove that $[n]$ attains the maximum number of rainbow 3-term arithmetic progression-free $r$-colorings among all subsets of $[n]$. Moreover, the exact number of rainbow 3-term arithmetic progression-free $r$-colorings of $\mathbb{Z}_p$ is obtained, where $p$ is any prime and $\mathbb{Z}_p$ is the cyclic group of order $p$.

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Xihe Li, Hajo Broersma, Ligong Wang. 2021-02-17. Integer colorings with no rainbow 3-term arithmetic progression. https://doi.org/10.37236/10249

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