All Questions
9 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
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
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): $...
- 4,605
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
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 ...
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