I am familiar with the recent Keevash paper here which proves that given some $t,n,k,\lambda$ then provided standard divisibility conditions hold, and $n$ is suitably large, there exists a $t-(n,k,\lambda)$ design.

My question is in a similar vein, given some $n,k,t$ with $n\geq 2k>2t$ (or something along those lines), does there always exist some $t-(n,k,\lambda)$ design? I don't suppose that there is a full answer, but I wonder if anyone knows any work done on this and if so, could you direct me towards it?

Many thanks!