# Tagged Questions

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vote

**1**answer

127 views

### Is there a “Bipartite” Szemeredi-Trotter theorem?

One version of the Szemeredi-Trotter theorem states the following:
Given a set of $L$ lines in the plane, the number of points incident to at least $k$ lines is bounded above by a constant times $L/k ...

**5**

votes

**1**answer

171 views

### Small remarkable matroids

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) ...

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**0**answers

31 views

### Generalizing Concepts of Planar Euclidean Geometry to Symmetric TSP-Instances

To me it seems possible, to successfully look at symmetric TSP instances from a geometry-point of view.
Examples are:
the diagonals of the convex hull of a set of points in the euclidean plane; ...

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votes

**0**answers

59 views

### Toric morphism fiber and kernel dimensions

Given a morphism between two smooth toric varieties $f: X \rightarrow Y$, is the dimension of the kernel of $\mathrm{d}f$ at any point $p \in X$ equal to the dimension of the fiber at $f(p) \in Y$?
...

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**2**answers

282 views

### When does a set of collinearity conditions imply collinearity of all of the points?

Suppose we have a set of $n$ points $\{X_1,X_2,\dots,X_n\}$ in the real plane and $\mathcal{A}$ a family of subsets of $\{1,\dots,n\}$.
By a "set of collinearity conditions for $\mathcal{A}$" we mean ...

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**1**answer

148 views

### Positroids and Totally Nonnegative Complex Grassmanian

Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case.
I found on arxiv the following interesting articles:
1)Alexander Postnikov: Total ...

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votes

**5**answers

414 views

### Lattice points in dilated polytopes and sumsets

Let $P$ be an integral polytope, that is, the convex hull of some points in $\mathbb{N}^d$.
Let $p_1,\dots,p_m$ be all lattice points in $P$.
Question: What is the condition on $P$ that guarantees ...

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vote

**0**answers

68 views

### Separating unit disks by circles

This is inspired by the recent question about separating unit disks by lines, which I will refer to as the "line case". Replacing "line" by "circle" adds one degree of freedom, and I'm wondering if ...

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**1**answer

225 views

### Separating unit disks by lines

Given $n\ge 2$. For a real $d>2$, consider a constellation $C$ of $2n$ disks of radius $1$ in the plane such that $h(C)$, the minimal distance between any two of their centers, is equal to $d$. Let ...

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**0**answers

300 views

### How many combinations does Android pattern have? [closed]

Rules-
1) At-least 4 and at-max 9 dots must be connected.
2) There can be no jumps
3) Once a dot is crossed, you can jump over it.

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votes

**1**answer

104 views

### How “accidental” are equalities between parts of Ehrhart quasi-polynomials? When do they persist to Euler-Maclaurin?

Background
What I think of Ehrhart theory (http://en.wikipedia.org/wiki/Ehrhart_polynomial) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th ...

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**0**answers

209 views

### Unique Domino Tiling

Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property?
Definitions:
A subset S of the xy-plane is star-convex if there is ...

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**1**answer

276 views

### Number of 3-tuple partitions of a multiple of three which follow the triangle inequality

Given n=3t, t$\in \mathbb N$; let $\mathbb L_3$ be set of all distinct integer partitions of n having 3 parts; say $\lambda_1,\lambda_2,\lambda_3$ .
If I chose any one partition randomly from ...

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**0**answers

74 views

### Group actions on polytopes in indefinite integer lattices

Is anything at all known about polytopes in indefinite integer lattices? I'm interested in lattice automorphisms which preserve certain polytopes of "high regularity" (e.g. cones). As a first step, ...

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votes

**1**answer

85 views

### Estimates on the number of vertices of reflexive polytopes

Suppose $M \cong \mathbb{Z}^n$ is a rank $n$ lattice, with dual lattice $N$. Suppose $\Delta$ is a full dimensional lattice polytope (i.e. convex hull of finite lattice points) in $M$. Then $\Delta$ ...

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**0**answers

219 views

### A problem in Galois Geometry

Given a prime $p$, out of $N$ vectors of length $p^k$ over $\Bbb F_2$ of Hamming weight $w^{k}$ that are chosen, how many vectors can there be with pairwise Hamming distance at least $2w^{k}$ given ...

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**2**answers

232 views

### Problems similar to Borsuk’s Theorem in the plane

Consider a 2-dimensional Borsuk's theorem:
Every bounded set $S$ in the plane can be partitioned into three parts with diameter smaller than the diameter of $S$.
I wonder if there are any ...

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votes

**2**answers

142 views

### Proving non-convexity of a set of lattice points

I have a set of lattice points S in R^n (listed in memory in a computer for n=8 say). I want to computationally certify that they do not form the lattice points of a convex polytope P in R^n. (Ex. ...

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votes

**1**answer

452 views

### Difference Sets

Suppose
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K
$$
We calculate the differences as: $$d=p_i-p_j\mod N,\quad i\ne j$$
Now let $a_d$ denote the number of occurrence of $d$ (for $d = 1, 2, \dots , N ...

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**0**answers

103 views

### What are the Voronoi cones in 4 variables?

Question: What are the top dimensional cones of the 2nd Voronoi decomposition of the space of positive definite forms in $4$ variables?
The 2nd Voronoi decomposition of the cone of positive definite ...

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**1**answer

265 views

### Maximum number of Vertices of Hypercube covered by Ball of radius R

Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...

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**1**answer

121 views

### Realizability of extensions of a free oriented matroid by an independent set

Question:
I am searching for a non-realizable matroid with few dependencies relative to the number of points. Precisely, I would like to find a non-realizable (over $\mathbb{R}$) oriented matroid $M$ ...

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218 views

### Perimeter/Neighborhood of a graph on grid

Hello,
I have a $\sqrt{n}\times\sqrt{n}$ lattice graph $G=(V,E)$ i.e. vertices on said 2-dim integer lattice, and two vertices have an edge if their $L_1$ distance is one.
Now I want to claim ...

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**2**answers

704 views

### Access to a preprint by D. N. Verma

Some work I am doing is connected with a sequence 1, 3, 40, 1225, 67956, $\dots$ which agrees with http://oeis.org/A012250 for all eight terms. The only useful information in OEIS on this sequence is ...

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**2**answers

551 views

### A Problem about partitioning $S^2$

Question: Can the 2-dimensional sphere $S^2$ be partitioned into four nonempty sets such that every circle in $S^2$ passes through just three of these four sets?
Here, "just three" means "exactly ...

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votes

**1**answer

641 views

### Sane bound on number of moves for Maker-Breaker game on $\mathbb R^2$ for $\{0,1,2,3,4\}$

The description below comes from
József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 ...

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**2**answers

225 views

### What is the smallest number of subsets in such a subdivision?

Given any $30$ points in the plane, what is the smallest number of
subsets in a subdivision of the set of $30$ points into subsets such
that all the points in each subset are on the boundary of the ...

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votes

**1**answer

214 views

### Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a
convex polygon in the plane and $v_{m+1}$ be a vertex in the interior
of the convex polygon. Connect ...

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votes

**2**answers

327 views

### On well separated point sets in the plane

Let us say that a finite set $A$ in the plane is $1$-separated if:
1) it has an even number of points;
2) no open ball of diameter $1$ contains more than $|A|/2$ points.
For a $1$-separated set $A$ ...

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vote

**0**answers

150 views

### Lattice-point enumeration question involving linear combinations of matrices

I would like to know some references to learn more about an answer to this question, if there are any references:
Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, ...

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**0**answers

304 views

### $n$ lines in a general position and the number of empty triangles

Question. Consider $n \geq 5$ lines in a general position (i.e. no two lines are parallel and no triple intersections are allowed) in $\mathbb{R}^2$. Let $T(n)$ denote the maximal number of empty ...

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331 views

### What collections of convex sets result in non-trivial uses of Helly's theorem?

Consider Helly Theorem, taken from notes by Igor Pak:
Let $X_1, \dots, X_n \in {\mathbb{R}}^2$ be convex regions in the plane such that any triple interesects $X_i \cap X_j \cap X_k \neq 0$. Then ...

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175 views

### Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold

Work by Tamura (extending results by Luo and Stong) shows the following.
Theorem: For any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for ...

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**1**answer

520 views

### Monomer-Dimer tatami tilings need better relationships with other math. Summary of results.

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...

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**1**answer

115 views

### Graph implemened into the plane with segments as edges and we search for matching with no edges intersecting

There are some points in the plane and some of them are connected with segments between them. We look at this structure as a graph implemented into the plane where the points are the vertices and the ...

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**1**answer

474 views

### Partitioning the vertices of an n-cube with random hyperplane cuts

An evolutionary biologist asked me a question which boils down, at least in part, to what seems to me an interesting question of combinatorial/probabilistic geometry.
It is an old chestnut of a ...

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vote

**1**answer

108 views

### Generalizing the internal angle of a graph in $\mathbb{E}^2$ to $\mathbb{S}^2$

I am currently working on research involving packing problems and am finding myself needing the tools from Combinatorial Geometry (in particular, I've been reading Pach and Agarwal's book on the ...

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**2**answers

2k views

### vector balancing problem

I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct.
This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest ...

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votes

**1**answer

338 views

### On the number of lines of given points

Hi all, I have a question Concerning Beck's theorem. I have read it from http://en.wikipedia.org/wiki/Beck%27s_theorem and I have two questions :
I suppose Beck's theorem doesn't hold when instead ...

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votes

**2**answers

463 views

### On the joints problem in finite fields

The original version of the so-called "joints problem" consists of the following:
Let $L$ be a set of lines in $\mathbb{R}^{3}$. Determine the maximum number of "joints" determined by these lines, ...

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vote

**1**answer

380 views

### Combinatorial optimization and graph coloring

I am considering the following problem:
(i) Fix $n$ and color the edges of $K_n$ red and blue arbitrarily.
(ii) Let $M$ be the set of monochromatic triangles in $K_n$ and define $g:M\rightarrow ...

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**0**answers

97 views

### Question regarding contiguous forms

I read about contiguous forms in Achill Scürmann's thesis on positive quadratic forms. I am wondering about one aspect of the Voronoi algorithm presented in there, that enumerates all arithmetically ...

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**0**answers

137 views

### Traversing perfect quadratic forms

I am wondering whether there is an efficient algorithm to traverse all the $N\times N$ perfect quadratic forms $Q$ inside the polyhedron $e_j^T Q e_j \geq 1$, $j = 1\ldots m$, where $e_j$ are some ...

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139 views

### Slices of Simplices that are Simplices, Reference?

I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.
Let $A$ be an $l\times n$ matrix with ...

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**3**answers

281 views

### Enumerating Perfect Lattices

I have a question regarding enumeration of perfect lattices/quadratic forms. In the thesis of Achill Schürmann http://fma2.math.uni-magdeburg.de/~achill/public/habil.pdf there is an algorithm called ...

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votes

**1**answer

537 views

### Why is the Street nerve of the Gray tensor product $[1]\otimes [1]$ isomorphic to $[1]\times [1]$

Recall that given two strict ω-categories $A$ and $B$, their lax Gray tensor product $A\otimes B$ is sent to the Verity-Gray tensor product of their associated complicial sets ...

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**8**answers

1k views

### Higher-dimensional Catalan numbers?

One could imagine defining various notions of higher-dimensional Catalan numbers,
by generalizing objects they count.
For example, because the Catalan numbers count the triangulations of convex ...

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vote

**1**answer

469 views

### On lattice points “far inside” convex lattice polygons

Let $\mathcal{P}$ be a convex lattice polygon with $n$ vertices and let $\mathcal{L}$ be the set of all lattice points inside $\mathcal{P}$. For every $n \geq 5$, does there exist a point in ...

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**3**answers

693 views

### Point sets in Euclidean space with a small number of distinct distances

It is well known and not hard to prove that the regular simplex in n-dimensions is the only way to place n+1 points so that the distance between distinct pairs of points is always the same. My general ...

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votes

**3**answers

623 views

### Not quite regular polyhedra

Take a naive interpretation of regular polyhedra:
All vertices (including epsilon ball) congruent
All edges congruent
All faces congruent
We can now find interesting families by removing one ...