31
votes

Accepted

### The Sylvester-Gallai theorem over $p$-adic fields

If $n \geq 3$ and $K$ is a field of characteristic not dividing $n$, containing a primitive $n$-th root of unity $\zeta$, then the $3n$ points of the form $(1:-\zeta^a:0)$, $(0:1:-\zeta^b)$, $(-\zeta^...

14
votes

Accepted

### Geometric interpretation of the exceptional isomorphism $PSp(4,3)=PSU(4,2^2)$

I believe that a finite geometric proof is given by Jean Dieudonné here:
Dieudonné, Jean, Les isomorphismes exceptionnels entre les groupes classiques finis, Can. J. Math. 6, 305-315 (1954). ZBL0055....

14
votes

Accepted

### About the paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl

The proof for this appeared over a series of papers. The final one was
Jan Saxl, `On Finite Linear Spaces with Almost Simple Flag-Transitive Automorphism Groups' Journal of Combinatorial Theory, ...

12
votes

### Is the sumset or the sumset of the square set always large?

Six years later it's hard to know if this is still a question of interest. For the record (since I stumbled across the question), the answer is yes and it's a consequence of a more general phenomenon.
...

12
votes

Accepted

### Can all lines in the euclidian plane be ordinary?

The answer is yes, by an argument using the axiom of choice.
There are exactly continuum many lines in the plane, and so by the
axiom of choice, we may enumerate them in a well-ordered sequence
of ...

11
votes

Accepted

### A vertical line with many intersections with $n$ non-parallel lines

For the solution see the attached figure here. It is the counterexample with $n=8$. Vertical lines represent all possible vertical lines which have less that $n$ intersection points with given lines.
...

10
votes

### The Sylvester-Gallai theorem over $p$-adic fields

Over any field $K$ of characteristic $\neq 2$ and containing $i := \sqrt{-1}$, there exists in $\mathbb{P}^2(K)$ a Sylvester-Gallai configuration with $12$ points given in affine coordinates by $(0,0)$...

10
votes

### Fano plane drawings: embedding PG(2,2) into the real plane

Here is a bit more symmetric version of the picture in the accepted answer:

10
votes

Accepted

### Which finite projective planes can have a symmetric incidence matrix?

The key word here is "polarity". A polarity of a projective plane with point set $P$ and line set $L$ is a map $\pi$ from $P \cup L$ to itself mapping points to lines and lines to points, ...

9
votes

Accepted

### Is every uniform hyperbolic linear space infinite?

The answer is "No".There exist a finite plane with required property.
First of all, let's rephrase your question. Your uniform linear space is a synonym to a BIBD (balanced incomplete block ...

8
votes

### Why should it be hard to generalize Dvir's proof of the finite field Kakeya conjecture to the Euclidean case?

I don't entirely follow your proposed approach, but let me make a few remarks which might help you think about this:
(1) Clearly this can't work, since this argument purports to show that a Kakeya ...

8
votes

Accepted

### Very symmetric quadrangle in $\Bbb CP^2$

That is true, since this is so for $\mathbb RP^n$ - take $n+2$ vertices of a "regular simplex" in it -- i.e. take the regular simplex in $S^n$ and project its vertices to $\mathbb RP^n$.
In ...

7
votes

### Blocking sets in three dimensional finite affine spaces

Here is a slightly better lower bound. If there are fewer than $2q^2$ points then there is some line that hits the set at most once. Consider the $q+1$ planes containing a line containing one point. ...

6
votes

### How many squares can be formed by $n$ points in general position in the plane?

We can get a lower bound on the order of $n \log n$.
I'll describe how to arrange $4^n$ points in general position to get $n 4^{n-1}$ squares.
The arrangement is described recursively. For the base ...

5
votes

### Point-line incidence bounds over positive characteristic fields

The current best Szemeredi-Trotter bound over finite fields also holds over arbitrary fields. It is in the wonderful paper "An Improved Point-Line Incidence Bound Over Arbitrary Fields" by ...

5
votes

Accepted

### Perfect matchings in infinite regular bipartite graphs

I believe this is correct (assuming $\lambda\gt0$).
If $\lambda$ is infinite then each connected component of $G$ has $\lambda$ vertices. Since the components can be handled independently, the problem ...

5
votes

### Was the small Desargues Theorem known to ancient Greeks?

This is more of a comment than an answer, but I think it deserves to be stated explicitly:
I don't know if the Greeks knew this result, but lest there be any doubt that the Greeks could have proved it,...

5
votes

Accepted

### Does every finite affine plane have the doubling property?

There is even stronger claim from which follows your answer.
Claim. Any affine plane obtained from Veblen-Weddenburn projective plane by dropping one line don't have "doubling" property.
...

4
votes

Accepted

### For which finite projective planes can the incidence structure be written as a circulant matrix?

To answer your questions:
1) A projective plane admits a circulant incidence matrix if and only if the automorphism group contains a cyclic group acting regularly on points and regularly on blocks. ...

4
votes

Accepted

### Lower bound on the distance set using incidences of points and circles

The Szemerédi–Trotter bound is known to be false for circles (it is true for circles with the same radii). There is a construction that gives $N^{2/3}|C|^{2/3}\log^{1/3}N$ incidences. The current best ...

4
votes

Accepted

### 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?

Take a 2-$(v,4,1)$ design on $v$ points and delete one block, along with the four points on it. In the original system each point is on exactly $(v-1)/3$ blocks, so if we assume $v\ge25$ the geometry ...

4
votes

### Applications of small Kakeya sets over finite fields

I became interested in Kakeya sets because they have the interesting property that a Kakeya set in a projective plane cannot be a subset of a blocking set, and with the exception of the full plane ...

4
votes

### Ree groups and Moufang octagons

Only having $k$ as subfield of $\ell$ is not sufficient (for both questions). The Ree groups (and the generalized octagons) are determined by a pair $(k, \theta)$, where $k$ is a field of ...

4
votes

### Is every uniform hyperbolic linear space infinite?

First answer was just existence of such space.
After routine computer search in large collection of BIBD collected by Vedran Krcadinac, I have two much smaller, nicer and more uniform examples.
First ...

4
votes

Accepted

### Is the group of translations of an affine plane always commutative?

Let $G$ be an arbitrary countably-infinite group. I claim that there is an affine plane $X$ such that $\operatorname{Trans}(X)$ contains a subgroup isomorphic to $G$.
By a back-and-forth argument one ...

3
votes

### Parallel lines containing a subset with even cardinality

It was conjectured by Dumitrescu and Toth in "Distinct triangle areas in a planar point set" (see Problem 1 at the end) that for every large enough $k$ for any $\mathscr P$ there is an ...

3
votes

Accepted

### Does any real projective plane incidence theorem follow from axioms?

This is false. I don't see how to get a counterexample from Andreas Blass's comment (the only uses for the existence of $\sqrt2$ which are obvious to me require a more flexible notion of incidence ...

3
votes

Accepted

### Injective choice function for finite Fano planes

Yes, there is always such a map. Let $k$ be the number of vertices in each edge of $H=(V,E)$. Consider an arbitrary vertex $v \in V$ and choose $e \in E$ such that $v \notin e$. For each $w \in e$ ...

3
votes

### Can one axiomatize projective lines using the cross-ratio?

Is the cross-ratio well-defined even when several of the four arguments coincide?
No, it is not. Modeled as homogeneous coordinates over $k^2$ you get the null vector in cases where three or more ...

3
votes

### Blocking sets in three dimensional finite affine spaces

Here is an improvement of the upper bound which I found in ``The polynomial method in Galois geometries'' by Simeon Ball. See page number 4.
The known constructions are somewhat crude. For example,...

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