For any s-cover on n points with lines of size at most (n−1)/(s−1), at least (n−1)/(s−1)+s−1 lines are needed, and this is tight when s−1 divides n−1.
Ray-Chaudhuri and Richard M
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Combining the theorems of Tur\'an and de Bruijn-Erd\H os
For any s-cover on n points with lines of size at most (n−1)/(s−1), at least (n−1)/(s−1)+s−1 lines are needed, and this is tight when s−1 divides n−1.