  Combining two classical notions in extremal combinatorics, the study of
Ramsey-Tur\'an theory seeks to determine, for integers $m\le n$ and $p \leq q$,
the number $\mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex
$K_q$-free graph in which every set of at least $m$ vertices contains a $K_p$.
  Two major open problems in this area from the 80s ask: (1) whether the
asymptotic extremal structure for the general case exhibits certain periodic
behaviour, resembling that of the special case when $p=2$; (2) constructing
analogues of Bollob\'as-Erd\H{o}s graphs with densities other than $1/2$.
  We refute the first conjecture by witnessing asymptotic extremal structures
that are drastically different from the $p=2$ case, and address the second
problem by constructing Bollob\'as-Erd\H{o}s-type graphs using high dimensional
complex spheres with all rational densities. Some matching upper bounds are
also provided.
