# Questions tagged [finite-geometry]

Galois geometry, finite projective and affine spaces, polar spaces, partial geometries, generalized polygons, near polygons, and other finite incidence geometries.

63 questions
Filter by
Sorted by
Tagged with
174 views

### Arrangement of subspaces over finite fields

I'm trying to find out what is already known about the following setup. Let $V$ be an $n$-dimensional vector space over a finite field $F_q$ (I'm mostly interested in the case where $q$ is prime), and ...
43 views

165 views

83 views

170 views

### Generalization of finite-projective-plane with more than one intersection point

In a finite projective plane, each two points appear together in exactly one line, and each two lines intersect in exactly one point. It is known that, if each line contains $n+1$ points, then the ...
200 views

### Status of the basis exchange condition for symplectic matroids

Let $J_n := \{1,2,3,\ldots,n,1^*,2^*,\ldots,n^*\}$ with the involution $x\mapsto x^*$ exchanging $i$ and $i^*$ for $1\leq i\leq n$. The following is supposed to be standard, but to avoid any doubt as ...
270 views

### Is there a well-known notion of orientability for finite geometries?

I'm wondering if the notion of an orientable/non-orientable manifold has any reasonable extension that allows for a similar classification of finite geometries. For example, the real projective plane ...
239 views

### Large gaps in Singer's difference sets

This question is related to the question I asked earlier. For a natural number $n$, a set $D$ of integer numbers is called a $n$-cyclic difference set if each integer number $x\notin n\mathbb Z$ can ...
350 views

### What is the smallest cardinality of a self-linked set in a finite cyclic group?

A subset $A$ of a group $G$ is defined to be self-linked if $A\cap gA\ne\emptyset$ for all $g\in G$. This happens if and only if $AA^{-1}=G$. For a finite group $G$ denote by $sl(G)$ the smallest ...
331 views

### Which cyclic groups admit a difference set?

Problem 1. For which $n$ does the cyclic group $C_n$ admit a difference set $D\subset C_n$, i.e., a set such that each non-unit element $x\in C_n$ can be uniquely written as the difference $x=ab^{-1}$...
239 views

### Large gaps in Singer planar difference sets?

By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C_n$. Such set $D$ is ...
108 views

### Is there literature on finite geometries with ordered lines?

A difference between finite geometries and (e.g.) Euclidean space is that "lines" in finite geometries are unordered subsets of the universe, while "lines" in Euclidean space are ordered subsets of ...
449 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 ...
869 views

### The exceptional isomorphism between PGL(3,2) and PSL(2,7): geometric origin?

It is well-known there is an isomorphism between $GL(3,2)=PGL(3,2)$, the automorphism group of the Fano plane (i.e. the projective plane over the finite field with two elements), and $PSL(2,7)$, which ...
224 views

### Maximum number of elements in union of subspaces

Let $V$ be a $m$-dimensional vector space over $\mathbb{F}_q$ and $1<\ell<m-1$. Let $r$ be a positive integer such that $r\ell\leq m$. QUESTION. What is the maximum number of elements in the ...
252 views

341 views

### $(n-2)$-blocking sets in $AG(n,2)$

Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$. I have seen a lot work related to minimal $(n-1)$-blockings set. ...
215 views

### An upper bound on the number of sets of parallel lines covering points in a finite plane?

Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a ...
146 views

### The Maximum Number of Lines Contained in the Point Set of a Finite Projective Plane

Consider a finite projective plane of order $q$. Define $f(m)$ to be the maximum number of lines completely contained in any point set of size $m$, where $1 \leq m \leq q^2+q+1$. I would like to ...
363 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 $q(q+1)/2 + (q-1)/2$ ...
225 views

### Sets of spreads in graphs

Let $G$ be a graph. A $k$-spread is a set of cliques of order $k$ which partition the vertex set (so $k|n$, where $n$ is the number of vertices). A partial $k$-resolution of $G$ is a set of pairwise ...
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 $\begin{pmatrix}1 & 1\\ 1& 1\end{pmatrix}$ ...
169 views