# Questions tagged [finite-geometry]

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### Finite nets, numerical invariants

I am studying the following paper: Finite Nets, Numerical Invariants (R. H. Bruck) for a school project about finite geometry. While I understand most of what is written, some points are rapidly ...
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### Example of a finite projective plane which is not a translation plane

I've been studying finite projective geometry for several weeks and I came across the fact that the most studied planes are the translation planes. Is there any known example (of minimum order if ...
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### Are there interesting finite geometries with angles or triangles which satisfy a Pythagorean-like identity?

I was playing around with Pythagoras's Theorem, and the proofs I was looking at involved areas and congruent triangles. I was wondering how much of this machinery we actually need, and especially if ...
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### If a finite group acts faithfully and primitively on the points of a configuration, will the figure be flag transitive?

Suppose we have a finite group G, and we had a configuration of points, lines etc. and lets say that the group acts faithfully and primitively on the set of points. Will the group be guaranteed to be ...
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1 vote
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### Maximal set of mutually skew lines in a finite projective space

What is the maximum size of a set of mutually-skew lines in a finite projective space? The total number of lines in a finite projective space is well-known ($(q^2 + 1)(q^2 + q + 1)$) and mentioned in ...
• 273
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### Are there parameters such that a combinatorial $(n_s,m_t)$ configuration does not exist?

It is well known that given a $(n_s,m_t)$ configuration the following must hold: $$ms=nt$$ $$s(t-1)+1\leq m$$ $$t(s-1)+1\leq n$$ However, for example, a $(43_7,43_7)$ configuration would be an order 6 ...
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### Are there ${n^2+n+1}_{n+1}$ configurations that are not projective planes?

Finite projective planes can be considered as combinatorial ${n^2+n+1}_{n+1}$ configurations. So for example the order 2 projective plane (Fano plane) is a $7_3$ configuration. It is known that the ...
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### Does every two-set partition of a projective plane contain a line?

Define a two-set partition of a projective plane as a partition of the points into two sets. Does there exist for any two-set partition a set in the partition that contains a line? What about infinite ...
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1 vote
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### Conditions for realizability of finite incidence structures in Euclidean plane

Given a finite incidence structure or equivalently a finite collinearity structure (satisfying standard set of axioms), I am interested in sufficient conditions for realizability in Euclidean plane (...
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### How many ways to arrange $n$ points in $(\Bbb F_q)^2$ with no three collinear?

How many ways are there to arrange $n$ points in the finite field plane $(\Bbb F_q)^2$ with no three of the points collinear? An easy upper bound is $(q^2)^n=q^{2n}$, but of course it's less than that....
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### Common divisor of Gaussian coefficient expressions

I have a question about common divisors of some expressions involving Gaussian coefficients, in particular in the case ${n \brack 1}_{q} = \frac{q^{n}-1}{q-1}$ where $q$ is a prime power. It is well ...
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1 vote
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### Lower bound of the Turan number $ex(n,C_4)$.

I want to show the lower bound $\Omega(n^{3/2})$ for the Turan number $ex(n,C_4)$. For a prime power $q$, a finite affine plane of order $q$ has $q^2$ points and $q^2 + q$ lines; each line contains $q$...
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### line at infinity intersected with a line from points of the parabola

Suppose we are looking at a parabola as a conic in $PG(2,\mathbb{K})$ with the line at infinity denoted $\ell_\infty$. I am still working on a previous problem I posted here. Suppose we have two ...
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### How many points belong to a line in a finite field?

This is the continuation of this question: What is the size of the set of lines in a finite field $\mathbb{F}_q$ of order $q$, where $q$ is a prime power? Now I need to show that exactly $q$ points ...
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### On some special 5-tuples in projective space $PG(3,2)$

Projective space $PG(3,2)$ has nice 5-tuples of points like $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$, $(1,1,1,1).$ This 5-tuple is "nice" because these points with their pairwise sums cover ...
1 vote
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### Lower bound on line blocking sets in the finite projective plane

I am looking at Bruen and Rotschild's article on lower bounds for line blocking sets in the finite projective plane of order $q$. In their proof on page 8, they use the summations \begin{align} \sum_{...
483 views

### Intuition behind lines and points in the projective plane

I've just started learning about projective plane and have trouble understand how to visualize the points and lines in the plane. Specifically, for this lemma below, when its says "all points of $l$ ...
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### how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$

I have a problem with the following question: how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$. How I tried to solve this problem. Here we have ...
254 views

### Conjugacy of Singer cyclic groups in $\mathrm{P\Gamma L}$

Motivation This is kind of a follow-up to this question on conjugacy of Singer cyclic groups in GL. The "original" definition of a Singer cycle is not in the GL, but the following slightly different ...
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### Is a perfect game of Set always possible?

For anyone not familiar with the game of Set, I'll refer you to the description on this question. My question is this: The game ends when there are no more cards remaining in the deck and there are ...
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### Primitive roots of unity occuring as eigenvalues of a product

I am currently trying to understand the proof of Benson's Lemma (1.9.1) in Generalized Quadrangles by Payne and Thas. Background We have two $k × k$ matrices $Q$ and $M$. We want to determine some ...
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I have a project to do on Generalised Quadrangles, specifically GQ(2,2). The project needs to have information about the construction of GQ(2,2), to prove this construction meets the conditions of a ...
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### Finite Incidence Geometry Questions

Definition of finite incidence geometry given: Consists of a finite set $P$ of points and a set of nonempty subsets of $P$ called lines, that satisfy the axioms: (F1) Two points determine exactly one ...
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For those familiar with combinatorial design, a projective plane is a $(q^2+q+1, q+1, 1)$ design. Geometrically, a projective plane of order $q$ is a set of points and lines through these points. ...