Bondarenkos counter-example in dimension $\geq 65$ to Borsuk's conjecture. Just to remind you, Borsuk conjectured that:

Every subset $E\subset \mathbb{R}^d$ can be partitioned into $(d+1)$ sets with smaller diameter.

Even though this conjecture had been proven to be wrong, the search for the smallest dimension in which it doesn't hold is still on.
After a couple of super high dimensional counter-examples Bondarenko showed in one of his papers that Borsuk's conjecture doesn't hold for dimension $64$. Which is, as far as i know, the current record.
To keep things short I skip some definitions on strongly-regular graphs. All necessary information is contained in the paper (see section on strongly-regular graphs).
Bondarenko uses a representation of strongly regular graphs to construct a two-distance set in a dimension. In detail they show that for the strongly-regular graph with parameters $G=(416,100,36,20)$ can be embedded into an $f=65$ dimensional space such that $84$-partitions are needed.
My question is: Why is $G=(416,100,36,20)$ the way to go? There are countless smaller strongly-regular graphs which can be embedded into smaller dimensions. How could they make sure that no smaller strongly-regular graph exists such that their proof works in smaller dimensions?
A list of (many) strongly-regular graphs can be found here.

First:
The considered paper by Bondarenko gave counterexamples for (all integer) dimensions from 65 onwards but not for a smaller dimension.
Second:
The mentioned arXiv reference is for the preprint. The final paper appeared in the journal Discrete & Computational Geometry, volume 51, issue 3 (April 2014). Meanwhile, it is freely accessible via
https://link.springer.com/article/10.1007/s00454-014-9579-4
Third:
Your question should be answered by Remark 1 of the final paper. (That remark states in particular, that the two graphs mentioned/used in the article had been found by an extensive search, and that the smaller one  "probably [...] has the smallest possible dimension for this approach. However, we can not prove this [...]").
