# 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).

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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[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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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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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 ...

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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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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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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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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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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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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 ...

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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 ...

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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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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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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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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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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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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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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 ...

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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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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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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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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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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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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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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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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,...

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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 $\mathfrak P$ ...

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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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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 $k$ ...

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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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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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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....