On sequences covering all rainbow k-progressions

Leonardo Alese, Stefan Lendl, Paul Tabatabai

Research output: Contribution to journalArticle

Abstract

Let $\ac(n,k)$ denote the smallest positive integer
with the property that there exists an $n$-colouring $f$ of
$\{1,\dots,\ac(n,k)\}$ such that for every $k$-subset
$R \subseteq \{1, \dots, n\}$ there exists an (arithmetic)
$k$\nobreakdash-progression $A$ in $\{1,\dots,\ac(n,k)\}$
with $\{f(a) : a \in A\} = R$.

Determining the behaviour of the function $\ac(n,k)$
is a previously unstudied problem.
We use the first moment method to give
an asymptotic upper bound for $\ac(n,k)$ for the case $k = o(n^{1/{5}})$.
Original language English Journal of Combinatorics 9 4 https://doi.org/10.4310/JOC.2018.v9.n4.a9 Published - 2018

Keywords

• rainbow arithmetic progression
• colouring
• covering
• arithmetic progression
• probabilistic method
• universal sequence