All Questions
Tagged with incidence-geometry discrete-geometry
17 questions
32
votes
0
answers
1k
views
Minimal number of intersections in a convex $n$-gon?
For a convex polygon $P$, draw all the diagonals of $P$ and consider the intersection points made by those diagonals. Let $f(n)$ be the minimal number of such intersections where $P$ ranges over all ...
- 1,474
31
votes
2
answers
1k
views
The Sylvester-Gallai theorem over $p$-adic fields
The famous Sylvester-Gallai theorem states that for any finite set $X$ of points in the plane $\mathbf{R}^2$, not all on a line, there is a line passing through exactly two points of $X$.
What ...
- 20.8k
16
votes
3
answers
2k
views
Fano plane drawings: embedding PG(2,2) into the real plane
By a drawing of the Fano plane I mean a system of seven simple curves and
seven points in the real plane such that
every point lies on exactly three curves, and every curve contains
exactly three ...
- 23k
8
votes
1
answer
246
views
A vertical line with many intersections with $n$ non-parallel lines
Pick $n\ge 3$ non-vertical lines $\mathscr{L}:=\{\ell_1,\ldots,\ell_n\}$ in the plane which are pairwise non-parallel, and they are not all concurrent in a single point.
Question. Does there exist a ...
- 1,511
7
votes
1
answer
436
views
When is a 0-1 matrix a one-intersection incidence matrix?
The following problem is what motivated my previous MO question.
It is easily seen that for any given 0-1 matrix $M$, one can always find
a set $\mathcal P$ of points, and a set $\mathcal C$ of simple ...
- 23k
7
votes
0
answers
123
views
Points on $k$ Circles
Let $k$ be a fixed positive integer. We want to find the minimum number $f(k)$, such that for a set of finite points in the plane, if any $f(k)$ of them are on $k$ circles, then all of them are on $k$ ...
- 628
6
votes
2
answers
992
views
On the joints problem in finite fields
The original version of the so-called "joints problem" consists of the following:
Let $L$ be a set of lines in $\mathbb{R}^{3}$. Determine the maximum number of "joints" determined by these lines, ...
- 625
5
votes
1
answer
436
views
How many squares can be formed by $n$ points in general position in the plane?
[This is much in the spirit (but different from) the questions from different posters: How many squares can be formed by using n points? and How many squares can be formed by using n points: revisited?...
- 13k
4
votes
1
answer
319
views
Planar sets closed under intersection of circles
Let $P$ be the plane with a point at infinity. By plane, I mean the Euclidian plane, and therefore it has circles. A line is also a circle, though its center is at infinity. If $A\subset P$ has ...
- 52.4k
4
votes
1
answer
561
views
Why should it be hard to generalize Dvir's proof of the finite field Kakeya conjecture to the Euclidean case?
Let $q$ be prime and let
$q\delta \sim 1.$ Let $K$ be any set of $C_n\delta$-separated tubes in $B(0,2)$, where $C_n$ is some constant depending on $n$. Let us consider a grid of $q^n$ points scaled ...
4
votes
0
answers
115
views
Projective planes over algebraically closed fields
Suppose I am given a projective plane $P \cong \mathbb{P}^2(k)$ over a (commutative) field $k$.
With "projective plane," I mean the point-line geometry (and not, for instance, the scheme): $...
- 4,595
4
votes
0
answers
443
views
Intersection of pencils in $\mathcal{R}^2$
Consider $9n$ pencils through non-collinear points $p_1, \ldots , p_{9n}$ in $R^2$ each consisting of at most $n$ concurrent lines. Define the intersection $S$ of these pencils to be the set of points ...
2
votes
1
answer
153
views
Very symmetric quadrangle in $\Bbb CP^2$
Is there a quadrangle $Q \subset \Bbb CP^2$, namely $Q$ is a set of four points, such that every permutation of $Q$ can be realizad by an isometric projectivity of $\Bbb CP^2$?
Clearly the analogous ...
- 2,283
2
votes
1
answer
205
views
Question involving an incidence geometry theorem from Larry Guth's book Polynomial Methods in Combinatorics [2016]
At the very beginning of Chapter 11 of Larry Guth's book, we are given the following theorem which is supposed to be proved within the chapter:
Theorem 11.1. There is a constant K so that the ...
2
votes
0
answers
157
views
A relation on triplets of points in the plane
This question is a follow up of my previous one (Planar sets closed under intersection of circles, Planar sets closed under intersection of circles) and is motivated by G. Zaimi's answer https://...
- 52.4k
1
vote
0
answers
129
views
The number of incidences between points and parabolas on $\mathbb{R}^2$
I was reading Adam Sheffer's book "Polynomial Methods and Incidence Theory" and I tried to solve the following exercise:
Exercise 1.1 Construct a set $\mathcal{P}$ of $m$ points and a set $\...
- 330
1
vote
1
answer
340
views
Is there a "Bipartite" Szemeredi-Trotter theorem?
One version of the Szemeredi-Trotter theorem states the following:
Given a set of $L$ lines in the plane, the number of points incident to at least $k$ lines is bounded above by a constant times $L/k ...
- 13