Define the Turán graph $T_{r}(n)$, where $r$ and $n$ are positive integers with $n \geqslant r$. For which $r$ and $n$ is $T_{r}(n)$ regular? For which $r$ and $n$ does $T_{r}(n)$ contain $T_{4}(8)$ as a subgraph?

State and prove Turán's theorem.

Let $x_{1}, \ldots, x_{n}$ be unit vectors in the plane. Prove that the number of pairs $i<j$ for which $x_{i}+x_{j}$ has length less than 1 is at most $\left\lfloor n^{2} / 4\right\rfloor$.