# Long rainbow arithmetic progressions

## TL;DR

This paper investigates the minimal number of colors needed to guarantee a rainbow arithmetic progression of length k in any equinumerous coloring, nearly closing the bounds with a new upper estimate.

## Contribution

The authors improve the upper bound on T_k, showing it is at most k^2 times a subexponential factor, narrowing the gap with the known lower bound.

## Key findings

- Established that T_k ≤ k^2 e^{(ln ln k)^2(1+o(1))}
- Confirmed the conjecture that T_k is close to quadratic in k
- Related work shows T_k=O(k^2 log k) independently.

## Abstract

Define $T_k$ as the minimal $t\in \mathbb{N}$ for which there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[tn]$ for all $n\in \mathbb{N}$. Jungi\'{c}, Licht (Fox), Mahdian, Nesetril and Radoici\'{c} proved that $\lfloor{\frac{k^2}{4}\rfloor}\le T_k$. We almost close the gap between the upper and lower bounds by proving that $T_k \le k^2e^{(\ln\ln k)^2(1+o(1))}$. Conlon, Fox and Sudakov have independently shown a stronger statement that $T_k=O(k^2\log k)$.