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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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Minimally intersecting subsets of fixed size

maybe you would be interested in the "Minimal Overlapping Algorithm" presented in "Two New Combinatorial Problems involving Dominating Sets for Lottery Schemes - Werner R Grundlingh" (at page 155): T …
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