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2
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0answers
19 views

Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
7
votes
0answers
81 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 $k$ ...
31
votes
2answers
600 views

How close can one get to the missing finite projective planes?

This question can be interpreted as an instance of the Zarankiewicz problem. Suppose we have an $n\times n$ matrix with entries in $\{0,1\}$ with no $\begin{pmatrix}1 & 1\\ 1& 1\end{pmatrix} $ ...
0
votes
0answers
119 views

“Open Points” in the 1983 proof of Szemerédi-Trotter theorem

I was reading through the 1983 paper "Extremal Problems in Discrete Geometry" and I was confused about the definition of "open point" appearing in this paper. By this point in the paper, the authors ...
1
vote
1answer
154 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 ...
1
vote
0answers
55 views

Point sets with tangents through every point

Let $D=(P,L)$ be either a $(v,k,\lambda)$-design or a near-linear space (or, more generally, any incidence structure with "points" and sets of points which are called "blocks" or "lines") and let $S ...
6
votes
1answer
265 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 ...
10
votes
1answer
368 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 ...
1
vote
2answers
120 views

Incidence matrices of generalized quadrangles

Is there somewhere a database of incidence matrices of generalized quadrangles that one can download?
2
votes
1answer
171 views

Is any $G$-set a coset geometry (in the sense of Tits-Buekenhout)?

Hi there! Let $X$ be a left $G$-set, and $\Delta=${$x_1,\ldots,x_n$} a fundamental domain of $G$ in $X$. In other words, $G$ acts on $X$ from the left, and {$Gx_1,\ldots,Gx_n$} is the orbit space ...
0
votes
1answer
214 views

About a graph embedding from R^3 to…

I was working on something and stumbled upon the following situation. I have in front of me a configuration $L$ of lines in $\mathbb{R}^{3}$ and say I consider the graph $G$ having as vertex set $L$ ...
2
votes
0answers
200 views

Axiomatization of the incidence geometry of the Euclidean plane

There are several well-known axiomatizations of Euclidean plane geometry, the language of which is usually considered to include at least the relations of incidence (point-line, point-segment, or ...
2
votes
0answers
132 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 ...
4
votes
1answer
290 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 ...
5
votes
2answers
536 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, ...
4
votes
0answers
417 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 ...
14
votes
2answers
966 views

Why do all incidence theorems follow from Pappus' theorem?

In Hilbert and Cohn-Vossen's ``Geometry and the Imagination," they state in the last paragraph of Chapter 20 that "Any theorems concerned solely with incidence relations in the [Euclidean ...
3
votes
1answer
467 views

What is the automorphism group of this geometry?

Define the following incidence structure of rank three. The points are the elements of $\mathbb{Z}_7=$ {$0,\ldots,6$}. The lines of type 1 are the triples $(x,x+1,x+3)$ modulo $7$. The lines of type 2 ...
3
votes
1answer
456 views

A question about the number of intersections of lines in $R^{3}$

Suppose I have n lines in $R^{3}$ with the conditions that: no 3 lines in one plane, no 3 lines intersect at one point, for fixed 2 lines, no 3 lines intersect these 2 lines at the same time. what is ...
2
votes
1answer
254 views

A rank 3 geometry for the sporadic simple group of Suzuki

Dear everyone, I am actually studying coset geometries (in the sense of Tits and Buekenhout) for the sporadic simple group of Suzuki. I came aware that Buekenhout found in 1979 a geometry over the ...
18
votes
1answer
1k views

A geometric series equalling a power of an integer

The following problem cropped up whilst considering generalised quadrangles with a product structure, and it boils down to a simple number theoretic problem. Let $s$ be an integer greater than 2 and ...
5
votes
2answers
675 views

Geometric interpretation of $BN$-pairs

My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres). ...
27
votes
2answers
2k views

Projective Plane of Order 12

I asked this question on the new Theoretical Computer Science "overflow" site, and commenters suggested I ask it here. That question is here, and it contains additional links, which I doubt I can ...