1
$\begingroup$

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

Please point out my mistakes and express your opinions!

Thank you very much!

$\endgroup$
4
  • 1
    $\begingroup$ Your link 1 does not resolve for me. What is the question? MO is generally not a place for stating conjectures without specific questions. $\endgroup$
    – LSpice
    Nov 29, 2021 at 3:50
  • $\begingroup$ Hi LSpice, I wanted to see if S(n)have upper limits '(n-b_m)'and 'n-b_m-d_m'. Do you have any suggestions? $\endgroup$
    – Scibee
    Nov 29, 2021 at 17:42
  • $\begingroup$ Have you seen Ben Green & Terence Tao, On Sets Defining Few Ordinary Lines, Discrete & Computational Geometry volume 50, pages409–468 (2013) ? $\endgroup$ Nov 30, 2021 at 9:58
  • $\begingroup$ They proved the number of ordinary lines is at least n/2. Furthermore, if n is odd number, the number of ordinary lines are at least 3n/4 - C, for some constant C $\endgroup$
    – Scibee
    Nov 30, 2021 at 18:28

0

Browse other questions tagged or ask your own question.