Contractibility of Vietoris-Rips Complexes of dense subsets in (Rⁿ, ₁) via hyperconvex embeddings

We consider the contractibility of Vietoris-Rips complexes of dense subsets of (Rⁿ,₁) with sufficiently large scales. This is motivated by a question by Matthew Zaremsky regarding whether for each n natural there is a rₙ>0 so that the Vietoris-Rips complex of (Zⁿ,₁) at scale r is contractible for all r≥ rₙ. We approach this question using results that relates to the neighborhood of embeddings into hyperconvex metric space of a metric space X and its connection to the Vietoris-Rips complex of X. In this manner, we provide positive answers to the question above for the case $n=2$ and $3$.