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The k-sets of S such that the subspaces S (d ≤ r—1) meet them at m or n points (m < n) are studied. Such sets are for d ≤ r—2 completely characterized from a numerical point of view.

This paper deals with the study of $\{k,n\}$-caps of a Galois space $S_{r,q}$ $(r\ge 3,q>2)$, that is of the sets of $k$ points of $S_{r,q}$ for which the maximum number of collinear points is $n$. Boundaries for $k$ ensuring that $\{k,n\}$-caps exist are given. Then we get a deeper insight of $\{k,n\}$-caps of the type $(m,n)$, that is, $\{k,n\}$-caps having only $m$-secant and $n$-secant lines. We prove in fact that, if such a cap exists and does not coincide with a hyperplane or with the complementary set of a hyperplane in $S_{r,q}$, then $q$ must be an odd square...