# Questions tagged [incidence-geometry]

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).

52
questions

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### graph built from orthogonal Latin Squares

I've asked the following question on MathExchange site, with a bounty, with no answer or comments. Maybe I would have additional comments here. The problem came to be while reading some articles on ...

**3**

votes

**1**answer

277 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 $Cn\delta$-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 ...

**6**

votes

**1**answer

142 views

### Does any real projective plane incidence theorem follow from axioms?

Is it known whether any projective geometry statement that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the standard projective axiomatics?
...

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votes

**1**answer

274 views

### Geometric interpretation of the exceptional isomorphism $PSp(4,3)=PSU(4,2^2)$

It is well-known that there is an isomorphism between $PSp(4,3)$ (the symplectic group of dimension $4$ over $\mathbb F_3$) and $PSU(4,2^2)$ (the unitary group defined by $4\times4$ unitary matrices ...

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votes

**1**answer

319 views

### About the paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl

The paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl called Linear spaces with flag transitive automorphism groups (Geom. Dedicata) from 1990 annonces a very powerful ...

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**1**answer

140 views

### Are two “perfectly dense” hypergraphs on $\mathbb{N}$ necessarily isomorphic?

We say that a hypergraph $(\mathbb{N}, E)$ where $E\subseteq {\cal P}(\mathbb N)$ is perfectly dense if
$\mathbb{N}\notin E$,
all $e\in E$ are infinite,
$e_1, e_2 \in E$ implies $|e_1\cap e_2| = 1$,...

**2**

votes

**0**answers

131 views

### theories where angles exist without a metric

The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted ...

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94 views

### What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?

Here are two facts that are superficially similar.
Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons.
Feit-Higman Theorem: The only ...

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**0**answers

180 views

### What are $(m,n)$-pseudoplanes?

An incidence geometry is a set $P$ (the "points"), a set $L$ (the "lines"), and a relation $I\subseteq P\times L$ ("incidence"). Equivalently, a bipartite graph with the halves of the partition ...

**2**

votes

**1**answer

136 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 ...

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**1**answer

149 views

### For which finite projective planes can the incidence structure be written as a circulant matrix?

It is well known that the projective plane of order $2$ can be represented by the circulant matrix $M_2:=circ(x,x,1,x,1,1,1)= \begin{pmatrix}
x&x&1&x&1&1&1\\
1&x&x&...

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88 views

### Bounds on k-tuple points for intersections of hyperplanes

Suppose that $H_1$,...,$H_d$ are hyperplanes in $\mathbb P^n$ (over some field -- you can pick). For $k \geq n$, let $t_k$ denote the number of points through which there pass exactly $k$ hyperplanes....

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81 views

### Lines meeting a given set in a unique point

Let $p$ be a fixed prime, and suppose that $S$ is a subset of the affine plane $\mathbb F_p^2$. If $|S|\le p+1$, then by the pigeonhole principle, through any given point $s\in S$ there is a line $L=L(...

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167 views

### On the determinant of incidence matrices (of graphs and other geometries)

Let $\Gamma = (P,L,I)$ be a point-line geometry (here, $P$ is the point set, $L$ the line set, and $I$ is the symmetric incidence relation). (As an example, $\Gamma$ could be a graph.) I suppose $\...

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**1**answer

145 views

### Lower bound on the distance set using incidences of points and circles

Suppose that $P$ is a set of $N$ points in the plane. Can we get a lower bound for the cardinality of the distance set $d(P)$ from the Szemerédi–Trotter theorem?
Here is my try.
The Szemerédi–Trotter ...

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votes

**3**answers

373 views

### Does there exist a finite hyperbolic geometry in which every line contains at least 3 points, but not every line contains the same number of points?

It seems to me that the answer should be yes, but my naive attempts to come up with an example have failed.
Just to clarify, by finite hyperbolic geometry I mean a finite set of points and lines such ...

**31**

votes

**2**answers

822 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 ...

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votes

**1**answer

371 views

### Can all lines in the euclidian plane be ordinary?

Is there a set $X \subset \mathbb{R}^2$ such that every straight line in the plane is ordinary in relation to it? i.e. if $r$ is any straight line then $|r \cap X|=2$.

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**1**answer

189 views

### Can one axiomatize projective lines using the cross-ratio?

I known axiomatizations of projective spaces of dimension > 2 and also of projective planes (either those obeying the axiom of Pappus, which come from fields, or those obeying the axiom of Desargues, ...

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votes

**2**answers

544 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 = 0,...

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votes

**4**answers

691 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 ...

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votes

**2**answers

505 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 (|\...

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votes

**0**answers

703 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 ...

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votes

**2**answers

948 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 $s_1,\dots,...

**9**

votes

**1**answer

360 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 $\mathfrak P$ ...

**6**

votes

**1**answer

342 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 $q(q+1)/2 + (q-1)/2$ ...

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votes

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115 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 $k$ ...

**38**

votes

**2**answers

1k 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} $ ...

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vote

**1**answer

228 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 ...

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votes

**0**answers

353 views

### Subplanes of Finite Projective Planes

If a finite projective plane $\pi_1$ of order $m$ contains, as a sub plane, a
finite projective plane $\pi_2$ of order $n$, then $m \geq n^2$ with equality holding only in the case of a Baer sub plane....

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65 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 \...

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votes

**1**answer

355 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 ...

**12**

votes

**2**answers

986 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 ...

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votes

**2**answers

204 views

### Incidence matrices of generalized quadrangles

Is there somewhere a database of incidence matrices of generalized quadrangles that one can download?

**2**

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**1**answer

222 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 $X/...

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**1**answer

339 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

**1**answer

274 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 $L$ ...

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votes

**0**answers

253 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 ...

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**0**answers

147 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://...

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votes

**1**answer

305 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 ...

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votes

**2**answers

855 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, ...

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votes

**0**answers

439 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 ...

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votes

**2**answers

496 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 ...

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**0**answers

519 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.
...

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votes

**2**answers

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 projective]...

**4**

votes

**1**answer

658 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 ...

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votes

**1**answer

480 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 ...

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votes

**1**answer

379 views

### A rank 3 geometry for the sporadic simple group of Suzuki

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 following diagram
...

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**1**answer

2k 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

**2**answers

785 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).
$[...