Combining two classical notions in extremal combinatorics, the study of Ramsey-Turán 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 Bollobás-Erdös 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 Bollobás-Erdös-type graphs using high dimensional complex spheres with \emph{all rational} densities. Some matching upper bounds are also provided.
This is joint work with Hong Liu, Christian Reiher and Katherine Staden.