Integer colorings with no rainbow 3-term arithmetic progression

Xihe Li; Hajo Broersma; Ligong Wang

doi:10.37236/10249

Integer colorings with no rainbow 3-term arithmetic progression

Xihe Li
, Hajo Broersma^*
, Ligong Wang

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

4 Citations (Scopus)

100 Downloads (Pure)

Abstract

In this paper, we study the rainbow Erdős-Rothschild problem with respect to 3term 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 Z_p is obtained, where p is any prime and Z_p is the cyclic group of order p.

Original language	English
Article number	P2.28
Journal	The Electronic journal of combinatorics
Volume	29
Issue number	2
DOIs	https://doi.org/10.37236/10249
Publication status	Published - 6 May 2022

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title = "Integer colorings with no rainbow 3-term arithmetic progression",

abstract = "In this paper, we study the rainbow Erd{\H o}s-Rothschild problem with respect to 3term 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 Zp is obtained, where p is any prime and Zp is the cyclic group of order p.",

author = "Xihe Li and Hajo Broersma and Ligong Wang",

note = "Funding Information: ∗Supported by the National Natural Science Foundation of China (No. arship Council (No. 201906290174). †Corresponding author. ‡Supported by the National Natural Science Foundation of China (No. Publisher Copyright: {\textcopyright} The authors.",

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AU - Li, Xihe

AU - Broersma, Hajo

AU - Wang, Ligong

N1 - Funding Information: ∗Supported by the National Natural Science Foundation of China (No. arship Council (No. 201906290174). †Corresponding author. ‡Supported by the National Natural Science Foundation of China (No. Publisher Copyright: © The authors.

PY - 2022/5/6

Y1 - 2022/5/6

N2 - In this paper, we study the rainbow Erdős-Rothschild problem with respect to 3term 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 Zp is obtained, where p is any prime and Zp is the cyclic group of order p.

AB - In this paper, we study the rainbow Erdős-Rothschild problem with respect to 3term 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 Zp is obtained, where p is any prime and Zp is the cyclic group of order p.

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DO - 10.37236/10249

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VL - 29

JO - The Electronic journal of combinatorics

JF - The Electronic journal of combinatorics

IS - 2

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ER -

Integer colorings with no rainbow 3-term arithmetic progression

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