Abstract incidence geometries like projective spaces, polar spaces, generalized polygons, as well as incidence problems in the real or complex Euclidean spaces (eg. Szemerédi–Trotter theorem).

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10
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275 views

Blocking sets in three dimensional finite affine spaces

What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line? Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = ...
14
votes
4answers
484 views

Synthetic projective lines

The classical synthetic notion of projective plane consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and ...
3
votes
2answers
306 views

Is the sumset or the sumset of the square set always large?

Let A be a finite subset of $\mathbb{N}$, $\mathbb{R}$, or a sufficiently small subset of $\mathbb{F}_{p}$. Do we have a lower bound of the form $|A|^{1+\delta}$ on the following quantity: $$\max ...
20
votes
0answers
313 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 ...
9
votes
2answers
899 views

Incidence geometry and matrices

Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines ...
9
votes
1answer
348 views

Generalized geometries

Let $S$ be a non-empty set. A geometry of type $n$ for $n\geq 1$ on $S$ (consisting of at least $n$ elements) is a set ${\mathfrak P}\subseteq {\mathcal P}(S)$ such that all members of $\mathfrak ...
4
votes
0answers
126 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
93 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 $k$ ...
34
votes
2answers
815 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} $ ...
1
vote
1answer
167 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 ...
2
votes
0answers
61 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 ...
7
votes
1answer
311 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 ...
11
votes
1answer
439 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 ...
3
votes
2answers
148 views

Incidence matrices of generalized quadrangles

Is there somewhere a database of incidence matrices of generalized quadrangles that one can download?
2
votes
1answer
184 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 ...
4
votes
1answer
219 views

Covering all, but $k$ points with affine subspaces

For non-negative integer $d\le n$ and $k\le 2^n$, how many affine subspaces of co-dimension $d$ are needed to cover all, but exactly $k$ elements of the vector space ${\mathbb F}_2^n$, and what are ...
0
votes
1answer
225 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
217 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
134 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
297 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
641 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
421 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 ...
4
votes
2answers
327 views

Reference on the Veblen-Young characterization of projective spaces

Can someone point me to a modern treatment of the Veblen-Young characterization of projective spaces of dimension greater than $2$ as $P(V)$ for some vector space $V$? [Added: see here for a ...
11
votes
0answers
441 views

Who conjectured that a transitive projective plane is Desarguesian?

The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved. ...
16
votes
2answers
1k 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 ...
4
votes
1answer
514 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
463 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
263 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
690 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 ...