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It is proved that a numerical semigroup can be associated to the triangle-free $(r,k)$-configurations, and some results on existence are deduced. For example it is proved that for any $r,k\geq 2$ there exists infinitely many $(r,k)$-configurations. Most proofs are given from a graph theoretical point of view, in the sense that the configurations are represented by their incidence graphs. An application to private information retrieval is described.

Associating a numerical semigroup to the triangle-free configurations