Geometric constructions for Ramsey-Turán theory

Combining two classical notions in extremal combinatorics, the study of Ramsey-Turán theory seeks to determine, for integers im/i ≤ in/i and ip/i ≤ iq/i, the number RTisubp/sub/i(in, Ksubq/sub, m/i) which is the maximum size of an in/i-vertex iKsubq/sub/i-free graph in which every set of at least im/i vertices contains a iKsubp/sub/i.br/brbr/br 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 ip/i=2; (2) constructing analogues of Bollobás-Erdős graphs with densities other than 1/2.br/brbr/br We refute the first conjecture by witnessing asymptotic extremal structures that are drastically different from the ip/i=2 case, and address the second problem by constructing Bollobás-Erdős-type graphs using high dimensional complex spheres with iall rational/i densities. Some matching upper bounds are also provided.