# Questions tagged [finite-geometry]

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

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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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### Non-Desarguesian finite projective planes with ≤3 (non-collinear) chosen points, and coordinatisation

It is well-known that an arbitrary projective plane can have very different symmetry group to a field plane. In particular, the symmetries are not transitive on the set of fundamental quadrangles. ...
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### 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 ...
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### Does every $C_4$-free bipartite graph lies in some finite projective plane?

A projective plane $Π$ is a 3-tuple $(P,L,I)$ where $P$ and $L$ are sets, and $I$ is a relation between $P$ and $L$, such that: For every two elements $p_1$, $p_2\in P$, there exists a unique ...
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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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### $\left< 15\right>^7/15$-womcode construction

In the article Womcodes constructed with projective geometries Frans Merkx constructed several good wom-codes (write-once memory codes, see How to reuse a "write-once" memory by Rivest & Shamir ...
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### Are bipartite Moore graphs Hamiltonian?

This is motivated by a computer-generated conjecture that bipartite distance-regular graphs are hamiltonian. I decided to check the case of Moore graphs first. The cycles and complete bipartite graphs ...
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### Almost blocking sets in $\mathbb F_q^2$

$\newcommand{\F}{{\mathbb F}}$ Let $q$ be an odd prime power. A blocking set in the affine plane $\F_q^2$ is a set blocking (meeting) every line. A union of two non-parallel lines is a blocking set ...
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### Largest number of points one can pick in finite projective space without getting three on a line

Consider the projectivization $\mathbb P\mathbb F_p^n$ of $\mathbb F_p^n$. How large a set $B \subseteq \mathbb P \mathbb F_p^n$ can I pick so that no three points of $B$ lie on the same line?
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### Applications of finite Bolyai-Lobachevsky planes

Google scholar gives more than 200 articles comcerning finite Bolyai-Lobachevsky (BL) planes. Usually they devoted to construction of such objects (axioms may be different). Are their any ...
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### Sets blocking every $2$-flat in $AG(n,2)$

The following may be well-known $-$ but not known to me: What is the smallest possible size of a set in ${\mathbb F}_2^n$ that blocks every $2$-flat? Here "blocks" means "have a non-empty ...
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### Finding a set of disjoint affine subspaces such that their union is equal to a given subset of $\mathbb{F}_2^n$

Suppose I'm given a set of point $S = \{x_1, \dots, x_m \} \subseteq \mathbb{F}_2^n$, and the following task. Find a set of disjoint affine subspaces of $\mathbb{F}_2^n$, $A_1, \dots, A_k$ satisfying ...
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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 $x = 0,... • 1,137 5 votes 1 answer 400 views ###$(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)$-blockings set. ... • 337 3 votes 1 answer 221 views ### An upper bound on the number of sets of parallel lines covering points in a finite plane? Let$\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 ... • 133 8 votes 0 answers 152 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 ... 6 votes 1 answer 397 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$q(q+1)/2 + (q-1)/2$... • 1,137 4 votes 0 answers 240 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 ... • 6,662 38 votes 2 answers 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} $... • 82.7k 2 votes 1 answer 173 views ### Is inner product preserved only by the stabiliser in a finite reflection group? Is the following statement true for finite reflection groups? Let$G$be a finite reflection group acting on$\mathbb{R}^n$, let$x, y\in \mathbb{R}^n$and let$z$be in the orbit of$y$. If$\...
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Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
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....