Extensions to the remaining parameter range and equality characterization

Determine whether analogous formulae for the chromatic extremal parameter hold in the remaining parameter range m<a<t, where the known constructions have a different form, and determine the full set of pairs (r,t) for which the equality f(n,r,t+1)=\delta(n,r,t) holds.

Background

The paper determines the chromatic extremal parameter \delta(n,r,t) in the range r=mt-a with 2\le a\le\min{m,t-1}, and uses this to resolve the balanced multipartite K_4-free minimum-degree problem for all admissible r\ge4 and n\ge1. The authors explicitly identify the complementary range m<a<t as unresolved because the known constructions in that range have a different form.

A second unresolved issue is to characterize all pairs (r,t) for which the multipartite (t+1)-clique-free threshold f(n,r,t+1) equals the chromatic threshold \delta(n,r,t).