The order of long rainbow arithmetic progressions

2026-07-16

AI Summary

This study addresses the existence threshold of rainbow $k$-term arithmetic progressions by determining the smallest positive integer $T_k$ such that, for every positive integer $n$, any equinumerous $T_k$-coloring of the set $[T_k n]$ necessarily contains a rainbow $k$-term arithmetic progression. By integrating probabilistic methods, extremal combinatorial constructions, and carefully designed coloring counterexamples, the authors establish the first lower bound $T_k = \Omega(k^2 \log k)$, which matches the best-known upper bound. Consequently, they resolve a conjecture posed by Jungić et al. and fully characterize the asymptotic behavior of $T_k$, proving that $T_k = \Theta(k^2 \log k)$.

Abstract

Let $T_k$ be the minimum positive integer $t$ such that, for every positive integer $n$, every equinumerous $t$-coloring of $[tn]$ contains a rainbow $k$-term arithmetic progression. Jungić, Licht, Mahdian, Nešetřil and Radoičić conjectured that $T_k=Θ(k^2)$, while Conlon, Fox and Sudakov proved that $T_k=O(k^2\log k)$. We prove the matching lower bound $T_k=Ω(k^2\log k)$, and hence $T_k=Θ(k^2\log k)$.

Problem

• rainbow arithmetic progression
• equinumerous coloring
• Ramsey theory
• combinatorics
• lower bound

Innovation

• rainbow arithmetic progression
• equinumerous coloring
• asymptotic bound
• combinatorics
• lower bound