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Design theory is the subfield of combinatorics concerning the existence and construction of highly symmetric arrangements. Finite projective planes, latin squares, and Steiner triple systems are examples of designs.

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Known result about existence of $n$-vertex $k$-uniform $r$-hypergraphs?

Are there known results about $n,k,r$ such that $n$-vertex $k$-uniform $r$-regular hypergraphs exist? If this is too large a class of hypergraphs, what if $k=\tilde{\theta}(\sqrt{n})$? What if an extr …