# 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 language English Journal of combinatorics 9 4 https://doi.org/10.4310/JOC.2018.v9.n4.a9 Published - 2018

Moment Method
Progression
Colouring
Covering
Upper bound
Denote
Integer
Subset

### Keywords

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

### Cite this

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

Research output: Contribution to journalArticleResearchpeer-review

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title = "On sequences covering all rainbow k-progressions",
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 givean asymptotic upper bound for $\ac(n,k)$ for the case $k = o(n^{1/{5}})$.",
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author = "Leonardo Alese and Stefan Lendl and Paul Tabatabai",
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TY - JOUR

T1 - On sequences covering all rainbow k-progressions

AU - Alese, Leonardo

AU - Lendl, Stefan

AU - Tabatabai, Paul

N1 - pp. 739-745

PY - 2018

Y1 - 2018

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

U2 - http://dx.doi.org/10.4310/JOC.2018.v9.n4.a9

DO - http://dx.doi.org/10.4310/JOC.2018.v9.n4.a9

M3 - Article

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JO - Journal of combinatorics

JF - Journal of combinatorics

SN - 2156-3527

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