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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!

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The best place to look for specific quadruples is the Handbook of Combinatorial Designs. If you want general statements, then one of Teirlinck's work (for large multiplicities), Kuperberg-Lovett-Peled (for large but usually more reasonable multiplicities) and Keevash's work are more or less all you can look for.

On the negative front, if you take the lines over a projective plane, then you have a design with multiplicity 1. It's easy to check (fix an edge and look at how the other edges are supposed to intersect it in some fixed uniformity-2 vertices) that you cannot have larger edges with multiplicity 1 unless the design is trivial (one edge), so this means you need a lot more than $n>2k$ in general. This is actually on the point of a famous open problem: when do designs with these extremal parameters exist? Here it is known that the `standard divisibility' conditions do not suffice!

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