Ramsey lower bounds for bounded degree hypergraphs

We prove that for all $k\ge 3$ and any integers $\mathrm{\Delta },n$ with $n\ge 2^{\mathrm{\Delta }},$ there exists a $k$-graph on $n$ vertices with maximum degree at most $\mathrm{\Delta }$ such that $r(H)\ge {\text{tw}}_{k-1}(c_k\mathrm{\Delta })\cdot n$ for some constant $c_k>0$, where ${\text{tw}}_k$ denotes the tower function. This makes the first progress toward a problem proposed by Conlon, Fox, and Sudakov (2009), who asked whether $r(H)\ge {\text{tw}}_k(c_k\mathrm{\Delta })\cdot n$ holds. Our proof relies on a novel construction of a $k$-graph on a growing number of vertices $n$ while keeping the maximum degree bounded by a fixed $\mathrm{\Delta }$.