On sequences covering all rainbow $k$-progressions

TL;DR

This paper investigates the minimal length of sequences that can be colored to cover all rainbow k-progressions for any k-subset, providing asymptotic bounds for small k relative to n.

Contribution

It introduces the function ac(n,k) for rainbow k-progressions and establishes an asymptotic upper bound for ac(n,k) when k grows slower than n^{1/5}.

Findings

Provides the first asymptotic upper bound for ac(n,k)

Uses the first moment method in combinatorial analysis

Addresses an unstudied problem in rainbow progression covering

Abstract

Let $\text{ac}(n,k)$ denote the smallest positive integer with the property that there exists an $n$-colouring $f$ of $\{1,\dots,\text{ac}(n,k)\}$ such that for every $k$-subset $R \subseteq \{1, \dots, n\}$ there exists an (arithmetic) $k$-progression $A$ in $\{1,\dots,\text{ac}(n,k)\}$ with $\{f(a) : a \in A\} = R$. Determining the behaviour of the function $\text{ac}(n,k)$ is a previously unstudied problem. We use the first moment method to give an asymptotic upper bound for $\text{ac}(n,k)$ for the case $k = o(n^{1/{5}})$.