For a point cloud $P$ with $n$ vertices is there a nice formula for the maximum number of points in a persistence diagram of the Vietoris-Rips complex on this point cloud?
Since in a VR complex a $1$-cycle must consist of $4$ distinct vertices at any filtration value of $\epsilon$ there should be $\binom{n}{4}$ cycles maximum. But of course this isn't actually true because if all sets of $4$ points make cycles then we will end up having to fill in many triangles as well.
Any thoughts on this would be much appreciated.
You will likely be interested in the paper "Extremal Betti Numbers of Vietoris–Rips Complexes" by Michael Goff, from DCG in 2011: https://link.springer.com/article/10.1007/s00454-010-9274-z. See in particular Section 3 for results on 1-dimensional homology, although higher-dimensional homology is also considered.
A clarifying comment is that I suspect you are asking about Vietoris-Rips complexes of a point cloud $P$ embedded in some Euclidean space $\mathbb{R}^d$. This is also the context considered in the Goff paper, and the dependence on $d$ is considered. One could also ask about Vietoris-Rips complexes of a point cloud $P$ embedded in an arbitrary metric space, which might be equivalent to asking what is the largest Betti number possible from an arbitrary clique simplicial complex with $n$ vertices.
While searching again for the above paper by Goff, I also ran across the following slides: https://plv.colorado.edu/dmoon/assets/docs/nzp.pdf
Best, Henry