I am currently trying to understand the construction of maximal graph which contains no $K_4$ and sub-linear number of independent points in the graph. The original paper [On a Ramsey–Turán type problem](https://doi.org/10.1016/0095-8956(76)90057-5), although ground-breaking, is very hard to parse. It also glosses over some crucial details and claims some really non-trivial identities. I am listing some of them below (but there are probably more questions one can ask). 

 - Is there a way to partition a hyper-sphere of dimension $n$ into $N$ equal parts and bounded diameters?
 - Aren't $C$ and $A$, as defined in the paper, very close to each other? Importantly, doesn't that imply that $\delta > 1$ (and that's it)?
- Can anyone explain the argument made to find the diameter of the spherical cap that, as claimed on top of page 3?

  [1]: https://moodle.iitb.ac.in/pluginfile.php/12182/mod_forum/attachment/6780/Erdos-Bollobas.%20Ramsey-Turan_problem.pdf?forcedownload=1