Let S(n) be the minimum number of ordinary lines determined over every set of n non-collinear points.
S(11)=6 S(14)=7
A=(3/7)*n
{B} = {B|3 ≤ B < n, B ∈ Odd prime}, {D} = {D|4 ≤ D <n, D ∈ Square number}
Let bₘ be the total number of elements of the finite Set {B} Let dₘ be the total number of elements of the finite Set {D}
Conjecture1: A ≤ S(n) ≤ n−bₘ {n|3 ≤ n ≤ +∞, n ∈ positive integer}?
Conjecture2: A ≤ S(n) ≤ n−bₘ−dₘ {n|6 ≤ n ≤ +∞, n ∈ even numbers}?
For example: if n = 11, there are three odd numbers (3, 5, 7) in {B}, then bₘ=3, (3/7) ∗11 ≤ S(11) ≤ 11−3
If n = 14, there are five odd numbers (3, 5, 7, 11, 13) in {B}, then bₘ=5; for 14>6, there are two square numbers (4, 9) in {D}, so dₘ=2, (3/7) ∗ 14 ≤ S(14) ≤ 14−5−2
Link 1: https://en.jinzhao.wiki/wiki/Sylvester%E2%80%93Gallai_theorem
Link 2: https://oeis.org/A003034
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