Let $P_k$ be the path on $k$ vertices and $\ell=\lfloor\frac{k-1}{2}\rfloor$. If $n\geq k\geq 5$
then is $\mathrm{AR}(n,P_k)$ equal to $\max\left(\binom{k-2}{2}+1,
\binom{\ell-1}{2}+(\ell-1)(n-\ell+1)+\epsilon\right)$where $\epsilon=1$ if $k$ is odd and
$\epsilon=2$ otherwise?
A proof of the formula for $\mathrm{AR}(n,P_k)$ for all $n\geq k\geq 5$ has been announced by
Yuan [Yu21].
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