This is in reference to the paper On a Ramsey-Turán Type Problem
On Page 3, the authors claim that the maximum number of independent points in the constructed graph is atmost equal to the area of the spherical cap $C$ with $\operatorname{diam}(C) = 2 - \frac{2 \epsilon}{3\sqrt{k}}$
I can't understand how the authors can claim this at all. I have been stuck on this problem for the past month. Any help is appreciated