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!


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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