On sequences covering all rainbow k-progressions

Leonardo Alese, Stefan Lendl, Paul Tabatabai

Research output: Contribution to journalArticleResearchpeer-review

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 languageEnglish
JournalJournal of combinatorics
Volume9
Issue number4
DOIs
Publication statusPublished - 2018

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Moment Method
Progression
Colouring
Covering
Upper bound
Denote
Integer
Subset

Keywords

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

Cite this

On sequences covering all rainbow k-progressions. / Alese, Leonardo; Lendl, Stefan; Tabatabai, Paul.

In: Journal of combinatorics, Vol. 9, No. 4, 2018.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Lendl, Stefan

AU - Tabatabai, Paul

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N2 - 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 givean asymptotic upper bound for $\ac(n,k)$ for the case $k = o(n^{1/{5}})$.

AB - 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 givean asymptotic upper bound for $\ac(n,k)$ for the case $k = o(n^{1/{5}})$.

KW - rainbow arithmetic progression

KW - colouring

KW - covering

KW - arithmetic progression

KW - probabilistic method

KW - universal sequence

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