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, 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.

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, 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?