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 $k = o(n^{1/{5}})$.
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 |
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Journal | Journal of Combinatorics |
Volume | 9 |
Issue number | 4 |
DOIs | |
Publication status | Published - 2018 |
Keywords
- rainbow arithmetic progression
- colouring
- covering
- arithmetic progression
- probabilistic method
- universal sequence