# Points on a Plane: Does n/2 Lines Exist?

• Dragonfall
In summary, there is a conjecture by G.A. Dirac that states for any arrangement of n points on a plane, not all on a line, there will always be a point with at least n/2 lines incident. However, this conjecture has exceptions for n = 11 and n = 14. There is also a similar conjecture stating that for any collection of n points on a plane, not all collinear, there will be at least n/2 lines containing exactly two points. This conjecture also has exceptions for n = 7 and n = 13 in the projective plane. Both of these conjectures have not been proven.
Dragonfall
If you put n points on a plane, not all on a line, is there always a point with at least n/2 lines incident? The lines in question are determined by the points themselves.

Dragonfall said:
If you put n points on a plane, not all on a line, is there always a point with at least n/2 lines incident? The lines in question are determined by the points themselves.

Hi Dragonfall!

If don't get it … n-1 lines are always incident at each original point;

or if you exclude the original n points, then eg it's not even true for a regular pentagon.

If 3 points are on a line, then you count it as one line, not 3. If every point "sees" every other point, then yes, each point has n-1 lines incident. However it's possible that some points are "blocked" by others. Take a pencil; there is a point with n-1 lines incident, but all others have only 2.

If you replace it with $$\lfloor n/2\rfloor$$, it still holds.

ah, I understand now …

this is a 1951 conjecture of G.A.Dirac (stepson of the Dirac ) "for any arrangements of n points, not all on a line, the maximum number of incident lines through a point can not be less than [n/2], and he claimed it is true for n ≤ 14."

He had a similar conjecture (see http://en.wikipedia.org/wiki/Sylvester–Gallai_theorem#The_existence_of_many_ordinary_lines") …
for any collection of n points, not all collinear, there exist at least n⁄2 lines containing exactly two points.

The first conjecture has 4 similar exceptions for n = 11 (in Tedjn's link), and the second has a very easy exception for n = 7 (and an exception for n = 13 in the projective plane).

Neither conjecture (even with those exceptions) has been proved.

Last edited by a moderator:
Ah, this explains why I've failed to prove it. Thanks.

## 1. What is the definition of "points on a plane"?

The term "points on a plane" refers to a set of points that lie on a two-dimensional surface, known as a plane. These points have two coordinates, typically denoted as (x,y), and are used to describe the location of a point on the plane.

## 2. What is the significance of n/2 lines in relation to points on a plane?

The equation n/2 represents the number of lines that can be drawn between a set of n points on a plane. This is known as the line-connecting problem, and it asks whether it is possible to draw lines between all points without any lines crossing or overlapping.

## 3. Can n/2 lines always be drawn between points on a plane?

The answer to this question depends on the arrangement of the points on the plane. If the points are randomly distributed, it is unlikely that n/2 lines can be drawn without any crossings. However, if the points are placed in a specific pattern, such as a regular grid, it is possible to draw n/2 lines without any crossings.

## 4. What is the relationship between n/2 lines and the number of points on a plane?

The number of lines that can be drawn between points on a plane depends on the number of points. If there are n points, there can be a maximum of n/2 lines without any crossings. This relationship is based on the fact that each point can be connected to n-1 other points, and each line connects two points.

## 5. What are some real-life applications of the line-connecting problem?

The line-connecting problem has applications in various fields, such as transportation, computer science, and biology. For example, in transportation, it can be used to determine the most efficient routes for a network of cities. In computer science, it can be applied to data visualization and graph theory. In biology, it can be used to study the structures and connections in the human brain.

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