On sequences covering all rainbow k-progressions

Leonardo Alese, Stefan Lendl, Paul Tabatabai

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
JournalJournal of Combinatorics
Issue number4
Publication statusPublished - 2018


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

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