On Ramsey number of Steiner systems

A $k$-uniform hypergraph $H$ is called a partial $(k,\mathrm{\ell })$-system if every set of $\mathrm{\ell }$ vertices of $V(H)$ is contained in at most one edge of $H$. We prove the existence of a partial $(k,k-1)$-system $H$ whose Ramsey number with $r\ge 4$ colors grows as a tower of height $k-1$.