A note on long rainbow arithmetic progressions

Abstract

Jungić et al (2003) defined $T_k$ as the minimal number $t\in \mathbb{N}$ such that there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[tn]$ for every $n\in \mathbb{N}$. They proved that for every $k\ge 3$, $\lfloor \frac{k^2}{4}\rfloor <T_k\le \frac{k(k-1{)}^2}{2}$ and conjectured that $T_k=\Theta (k^2)$. We prove for all $\epsilon >0$ that $T_k=O(k^{5/2+\epsilon })$ using the Kővári-Sós-Turán theorem and Wigert's bound on the divisor function.

3 pages

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• Independent arithmetic progressions