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).

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9 votes
1 answer
298 views

Is the group of translations of an affine plane always commutative?

$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, ...
11 votes
1 answer
370 views

Does every finite affine plane have the doubling property?

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
3 votes
1 answer
201 views

Another implication of the Affine Desargues Axiom

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
9 votes
2 answers
361 views

Does the Affine Pappus Axiom imply the Affine Desargues Axiom in affine planes?

I am interested in the affine version of the well-known Hessenberg's Theorem (saying that Pappian projective planes are Desarguesian). First I introduce all necessary definitions. Definition L. A ...
5 votes
5 answers
545 views

Is every uniform hyperbolic linear space infinite?

I start with definitions. Definition 1. A linear space is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ satisfying three axioms: (L1) for any distinct ...
2 votes
1 answer
94 views

Is every Cartesian biaffine plane affine?

This question concerns the (synthetic) geometry of linear spaces. Definition 1. A linear space is a pair $(P,\mathcal L)$ consisting of a set $P$ whose elements are called points and a family $\...
2 votes
0 answers
152 views

A direct proof that every projectivity between parallel lines is affine

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
11 votes
3 answers
538 views

Was the small Desargues Theorem known to ancient Greeks?

My question concerns the classical Desargues Theorem and its simplest version The small Desargues Theorem: Let $A$, $B$, $C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...
7 votes
1 answer
324 views

A corollary of the affine Desargues axiom

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
41 votes
2 answers
5k 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 ...
1 vote
0 answers
119 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 $\...
1 vote
1 answer
277 views

Szemerédi–Trotter type theorem in finite field

This question is about the content of this paper by J. Bourgain, N. Katz, T. Tao. In the final step (page 18) of the proof of Szemerédi-Trotter type theorem, we have already known $$|A''+A''|\lesssim ...
2 votes
1 answer
116 views

Ree groups and Moufang octagons

Consider a Ree group of type $^2\mathrm{F}_4$, defined over the field $k$. Tits showed that every Moufang generalized octagon arises as a natural geometric module on which a Ree group of this type ...
0 votes
0 answers
88 views

What is $(C, D, \delta, \gamma)$ and $(C, \delta; D, \gamma)$ Desarguesian?

A projective plane is $(C, \gamma)$-Desarguesian if for any 2 triangles $A_1 B_1 C_1, A_2 B_2 C_2$ in perspective from $C$ (which means $C \in A_1 A_2, B_1 B_2, C_1 C_2$) such that $A_1 B_1 \cap A_2 ...
4 votes
1 answer
544 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 ...
5 votes
1 answer
304 views

Which finite projective planes can have a symmetric incidence matrix?

As the title says. Which finite projective planes admit a symmetric incidence matrix? I am not an expert in the field at all, but I consulted with one. He claimed that $PG(2, \mathbb F_q)$ can always ...
3 votes
0 answers
34 views

Baer involutions fixing the same plane

Let $\mathbf{PG}(2,q^2)$ be the finite projective plane defined over the finite field $\mathbb{F}_{q^2}$. Then for each quadrangle, there is precisely one involution fixing it pointwise, and hence ...
2 votes
0 answers
54 views

Classification of Moufang planes of real dimension 16

Incidence geometry is not really area of expertise so I'm asking here: are all Moufang planes of 16 dimension already classified? I'm not just interested in the compact ones. Is there already a ...
8 votes
1 answer
237 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 ...
3 votes
0 answers
36 views

Anti-flag transitive affine planes

Let $\mathcal{A}$ be an axiomatic affine plane. First let $\mathcal{A}$ be finite. Suppose that the automorphism group of $\mathcal{A}$ acts transitively on nonincident point-line pairs (that is, on ...
2 votes
0 answers
74 views

Anti-flag transitive projective planes

Let $\Gamma$ be an axiomatic projective plane, and suppose its automorphism group acts transitively on the anti-flags (the point-line pairs $(u,V)$ such that $u$ is not incident with $V$). In the ...
5 votes
1 answer
153 views

Parallel lines containing a subset with even cardinality

For each $\alpha \in \mathbf{R}\cup \{\infty\}$, let $\mathscr{L}_\alpha$ denote the collection of lines $\ell$ of $\mathbf{R}^2$ with slope $\alpha$. More explicitly: if $\alpha \in \mathbf{R}$, then ...
16 votes
3 answers
1k 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 ...
10 votes
1 answer
485 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....
4 votes
0 answers
111 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): $...
1 vote
0 answers
121 views

Combinatorics of projective planes over commutative rings

An axiomatic projective plane is a point-line incidence structure with the following axioms: any two distinct points are collinear (via a unique line); any two distinct lines meet in a unique point; ...
3 votes
3 answers
724 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 (|\...
2 votes
0 answers
90 views

Segre's theorem in $3$ dimensions with a "twist"

As I understand, there is a $3$-dimensional analogue of Segre's theorem stating that the maximum size of a set in ${\bf F}_q^3$ ($q$ odd) with no three points collinear is $q^2+1$. I am trying to ...
10 votes
0 answers
224 views

Projective planes over non-division rings

Is there a "right" notion of a projective plane over a general (unital, non-division) ring? Let me explain what type of object I am looking for. Let $R$ be an arbitrary (not necessarily ...
6 votes
1 answer
158 views

Point-line incidence bounds over positive characteristic fields

I am aware of work on point-line incidence bounds over $\mathbb{R}$, $\mathbb{C}$, and finite fields, in particular various versions of the Szemeredi-Trotter bounds. I would like to know if work along ...
3 votes
0 answers
78 views

Infinite-dimensional quasifields

In their seminal paper on translation planes (The Construction of Translation Planes from Projective Spaces, Journal of Algebra 1:85-102, 1964, https://doi.org/10.1016/0021-8693(64)90010-9), Bruck and ...
3 votes
1 answer
262 views

Perfect matchings in infinite regular bipartite graphs

This question was motivated by a discussion here and is related to a previous question here. Let $\kappa$ and $\lambda$ be cardinals such that $0<\lambda\leq \kappa$. Let $G=(A\cup B, E)$ be a ...
3 votes
1 answer
74 views

Injective choice function for finite Fano planes

Let $H=(V,E)$ be a hypergraph that is a finite Fano plane, that is, $V$ is a finite set and $E$ has the following properties: for $e_1\neq e_2\in E$ we have $|e_1|=|e_2|$, as well as $|e_1\cap e_2|=1$...
1 vote
0 answers
99 views

What can be said about a class of incidence structures closed under duals and complements?

Note that I do not work in combinatorics, and so this question might be a bit naive. The question is inspired by some structures that arise in my research within representation theory. Recall that an ...
2 votes
1 answer
184 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 ...
3 votes
0 answers
131 views

Generic linear subspaces of symmetric matrices

Let $\mathcal{S}_{n}(\mathbb{R})$ be the real vector space of symmetric $n\times n$ traceless matrices with real entries and let $L\subset \mathcal{S}_{n}(\mathbb{R})$ be a linear subspace. Noticing ...
5 votes
1 answer
414 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?...
2 votes
1 answer
139 views

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 ...
6 votes
1 answer
283 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? ...
16 votes
1 answer
383 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 ...
10 votes
1 answer
373 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 ...
4 votes
1 answer
153 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 $|e_1\cap e_2| = 1$,...
7 votes
0 answers
270 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
0 answers
155 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 ...
5 votes
0 answers
115 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 finite ...
2 votes
1 answer
151 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 ...
6 votes
4 answers
1k views

Is the theory of incidence geometry complete?

Consider the basic axioms of planar incidence geometry, which allow us to speak of in-betweeness, collinearity and concurrency. These axioms per se are not complete, since for example, Desargues ...
2 votes
1 answer
273 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&...
4 votes
0 answers
100 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....
6 votes
1 answer
442 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 $q(q+1)/2 + (q-1)/2$ ...