The maximal number of $3$-term arithmetic progressions in finite sets in different geometries

Green and Sisask showed that the maximal number of $3$-term arithmetic progressions in $n$-element sets of integers is $\lceil n^2/2\rceil$; it is easy to see that the same holds if the set of integers is replaced by the real line or by any Euclidean space. We study this problem in general metric spaces, where a triple $(a,b,c)$ of points in a metric space is considered a $3$-term arithmetic progression if $d(a,b)=d(b,c)=\frac{1}{2}d(a,c)$. In particular, we show that the result of Green and Sisask extends to any Cartan--Hadamard manifold (in particular, to the hyperbolic spaces), but does not hold in spherical geometry or in the $r$-regular tree, for any $r\geq 3$.