Obtain linear dependence on the maximum degree

Establish a lower-bound construction for Ramsey numbers of bounded-degree k-uniform hypergraphs whose dependence on the maximum degree Δ is linear on top of the tower, improving the established dependence from tw_k(c_k√Δ) to a tower with a linear-in-Δ top parameter.

Background

The main theorem constructs, for every k ≥ 2 and n ≥ Δ, an n-vertex k-uniform hypergraph of maximum degree at most Δ whose four-color Ramsey number is at least tw_k(c_k√Δ)·n. The tower height is optimal, but the top parameter has only square-root dependence on Δ. The authors explicitly identify obtaining linear dependence on Δ at the top of the tower as an unresolved improvement; the construction also requires four colors because it relies on the stepping-up lemma.