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).
90 questions
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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 ...
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Is every finite plane with a characteristic Desarguesian?
By a projective plane I understand a mathematical structure $(X,\mathcal L)$ consisting of a set $X$ of points and a family $\mathcal L$ of subsets of $X$, called lines such that the following four ...
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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).
$[...
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Exceptional Lenz-Barlotti classes IVa.3 and IVb.3
On this web-site, devoted to the Lenz-Barlotti classification of projective planes, it is written that the class IVa.3 (and its dual IVb.3) is somewhat exceptional, because it contains exactly one ...
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Lexicographically largest incidence matrix
I have simple algorithmic question, but I can't find any source where this algorithm is explained in details.
Let's assume that we have incidence (with 0 and 1 values) matrix of size $m\times n$. Let ...
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An algebraic characterization of dual translation projective planes
It is well-known that translation projective planes are coordinatized by quasifields. More precisely, a projective plane is translation if and only if it has a ternary-ring $R$ which is linear, the ...
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Can any properties of a ring other than being a field be captured by the geometry of its 2-dimensional free module?
Can any properties of a ring other than being a field be captured by the geometry of its 2-dimensional free module?
Background:
In his wonderful, wonderful book Geometric Algebra, Emil Artin describes ...
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Are quadruples $abcd$ and $dcba$ always projectively equivalent in any projective plane?
It is well-known that for every line $L$ in a Pappian projective plane (i.e., a projective plane over a field) and any distinct points $a,b,c,d\in L$ the quadruples $(a,b,c,d)$ and $(d,c,b,a)$ are ...
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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]...
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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, ...
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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\...
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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\...
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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 ...
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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 $\...
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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\...
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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 $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...
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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\...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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): $...
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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;
...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $|e_1\cap e_2|=1$...
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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 ...
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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 ...
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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 ...
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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?...
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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 ...
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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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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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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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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 $|e_1\cap e_2| = 1$,...
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