I am familiar with Keevash's proof that provided divisibility conditions hold, $t-(n,k,\lambda)$ designs exist for all but finitely many $t,n,k,\lambda$.
My question is, given some $n,t,k$ does there exist a $t-(n,k,\lambda)$ design for some $\lambda$? I can't find any literature on this at all, and it seems unlikely to have an easy answer.
In particular, I would like to know in particular about the existence of $t-(n,t+1,\lambda)$ designs.
Thanks all!