# 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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### 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 ...
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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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### Is there a hyperplane avoiding two independent sets?

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....