Integer colorings with no rainbow 3-term arithmetic progression

Abstract

In this paper, we study the rainbow Erdős-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 $p$.

arXiv comment13 pages

References in corpus (8)

• Hypergraph containers
• On the structure of oriented graphs and digraphs with forbidden tournaments or cycles
• The typical structure of Gallai colorings and their extremal graphs
• On the maximum number of integer colourings with forbidden monochromatic sums
• The number of Gallai k-colorings of complete graphs
• Integer colorings with forbidden rainbow sums
• Anti-van der Waerden numbers of 3-term arithmetic progressions
• The number of subsets of integers with no $k$-term arithmetic progression