Geometric constructions for Ramsey–Turán theory

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) how to construct analogues of Bollobás–Erdős graphs with densities other than \frac{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–Erdős-type graphs using high-dimensional complex spheres with all rational densities. Some matching upper bounds are also provided.