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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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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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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Incidence matrices of generalized quadrangles
Is there somewhere a database of incidence matrices of generalized quadrangles that one can download?
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
user1448
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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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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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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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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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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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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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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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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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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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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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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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