Geometric constructions for Ramsey-Tur\'an theory

Combining two classical notions in extremal combinatorics, the study of Ramsey-Tur'an theory seeks to determine, for integers $m\le n$ and $p\le q$, the number ${\mathsf{R}\mathsf{T}}_p(n,K_q,m)$, 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 $1\mathrm{/}2$. 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.