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Apr 2, 2019 at 16:17 comment added Gerhard Paseman Finally, I suspect the following will be optimal: take a projective plane where every point is on at most r lines, and then extend as needed to have each line contain k points. This gives something like O(r^2) for your f, and all that remains after proving the above optimal is to consider when k is smaller than r, and see how sharp the naive bounds are. Gerhard "Can't Get Out Of Suburb" Paseman, 2019.04.02.
Apr 2, 2019 at 15:16 comment added Gerhard Paseman There is also the simple upper bound k*(r-1)+1. Is this the restriction that really interests you? Gerhard "The Neighborhood Seems Very Small" Paseman, 2019.04.02.
Apr 2, 2019 at 15:12 comment added Gerhard Paseman You now have n*r/k as an upper bound, which likely is strong for small r. Is there a region of r that interests you? Gerhard "Is In The Same Neighborhood?" Paseman, 2019.04.02.
Apr 2, 2019 at 11:36 history asked Chris H CC BY-SA 4.0