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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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Has this kind of design been studied before?

Consider a design $(X,\mathcal{B})$, satisfying: Each block in $\mathcal{B}$ has the same size The intersection of every two blocks has the same size Of course, it is easy to find many examples of …