Close the two-color hypergraph Ramsey gap

Determine the correct growth rate of the two-color Ramsey number of the k-uniform clique K_n^{(k)} for k ≥ 3 by closing the gap between the known lower bound tw_{k-1}(Ω_k(n^2)) and the known upper bound tw_k(O_2(n)).

Background

For k-uniform hypergraph cliques with k ≥ 3, the paper reports a stepping-up lower bound of tw_{k-1}(Ω_k(n^2)) for two colors, while Erdős–Rado provide an upper bound of tw_k(O_2(n)). The authors note that the discrepancy between these bounds remains a major unresolved problem. This question concerns the classical dense hypergraph Ramsey setting and is distinct from the bounded-degree hypergraph constructions established in the paper.