A set $S$ of $q$-subsets of an $n$-set $X$ is a design with parameters $(n,q,r,\lambda)$ if every $r$-subset of $X$ belongs to exactly $\lambda$ elements of $S$. In other words, a design with parameters $(n,q,r,\lambda)$ is an $n$-vertex $q$-uniform hypergraph in which every $r$-subset of the vertex set belongs to exactly $\lambda$ edges. The existence of a design with parameters $(n,q,r,\lambda)$ is equivalent to a $K_q^r$-decomposition of $\lambda K_n^r$ (the complete $\lambda$-fold $r$-uniform hypergraph of order $n$). By Keevash's Theorem (2014), $\lambda K_n^r$ can be decomposed into $ K_q^r$ when some obvious divisibility conditions are satisfied and $n$ is sufficiently large. In this talk, I will discuss a ``multipartite" version of Keevash's Theorem.
Keywords: hypergraphs, designs, generalized designs, multipartite, amalgamation, detachment