All Questions
Tagged with design-theory or combinatorial-designs
21 questions
56
votes
21
answers
14k
views
Linear algebra proofs in combinatorics?
Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...
39
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} $ ...
- 85.6k
5
votes
1
answer
2k
views
What is the largest number of k-element subsets of a given n-element set S such that…
Given a set S of n elements. What is the largest number of k-element subsets of S such that every pair of these subsets has at most one common element?
23
votes
2
answers
3k
views
Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices?
The following problem is homework of a sort -- but homework I can't do!
The following problem is in Problem 1.F in Van Lint and Wilson:
Let $G$ be a graph where every vertex
has degree $d$. ...
- 156k
16
votes
3
answers
2k
views
Fano plane drawings: embedding PG(2,2) into the real plane
By a drawing of the Fano plane I mean a system of seven simple curves and
seven points in the real plane such that
every point lies on exactly three curves, and every curve contains
exactly three ...
- 23k
12
votes
3
answers
4k
views
Status of Hadamard matrix conjecture
I would like to know if any progress has been made on Hadamard conjecture :
Hadamard matrix of order $4k$ exists for every positive integer $k$.
- 121
12
votes
4
answers
3k
views
What are the major open problems in design theory nowaday?
I gather that the question whether the Bruck-Chowla-Ryser condition was sufficient used to top the list, but now that that's settled - what is considered the most interesting open question?
11
votes
4
answers
5k
views
Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices
What's the maximum determinant of $\{0,1\}$ matrices in $M(n,\mathbb{R})$?
If there's no exact formula what are the nearest upper and lower bounds do you know?
- 111
10
votes
1
answer
319
views
Construction of skew-Hadamard matrix of order 292
I am currently looking into how to construct a skew-Hadamard matrix of order 292. Where can I find such construction?
According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard ...
- 261
10
votes
2
answers
632
views
Seeking very regular $\mathbb Q$-acyclic complexes
This question was raised from a project with Nati Linial and Yuval Peled
We are seeking a $3$-dimensional simplicial complex $K$ on $12$ vertices with the following properties
a) $K$ has a complete $...
- 24.7k
8
votes
0
answers
210
views
More about self-complementary block designs
For what odd integers $n \geq 3$ does there exist a self-complementary $(2n,8n−4,4n−2,n,2n−2)$ balanced incomplete block design?
By "self-complementary" I mean that the complement of each block is a ...
- 19.7k
7
votes
2
answers
307
views
Self-complementary block designs
For what $n$ does there exist a self-complementary
$(2n,4n-2,2n-1,n,n-1)$ balanced incomplete block design?
(All I know is that a self-complementary design with these parameters does exist for all $...
- 19.7k
6
votes
1
answer
151
views
How to construct a skew Hadamard matrix of order 756?
Where can I find the construction for a skew Hadamard matrix of order 756?
According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard matrices and Seberry - On skew Hadamard ...
- 261
6
votes
3
answers
678
views
Meeting management
A friend wants to have ten meetings of six people every day for five days with no pair of people meeting twice. Is this possible? It appears to be a question about maximal decomposition of a complete ...
- 69
5
votes
0
answers
911
views
The existence of big incompatible families of weight supports
In 2018 Mario Krenn posed this originated from recent advances in quantum physics question on a maximum number of colors of a monochromatic graph with $n$ vertices. Despite very intensive Krenn’s ...
- 5,409
5
votes
5
answers
572
views
Is every uniform hyperbolic linear space infinite?
I start with definitions.
Definition 1. A linear space is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ satisfying three axioms:
(L1) for any distinct ...
- 41.8k
4
votes
3
answers
780
views
Does an $(x, bx)$-biregular graph always contain a $x$-regular bipartite subgraph?
I guess a discrete-mathematics-related question is still welcome in MO since I was new to the community and learned from this amazing past post. The following claim is a simplified and abstract form ...
- 43
4
votes
1
answer
1k
views
Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and $q$-binomials?
The short version of my question is:
1)For which positive integers $k, n$ is there a solution to the equation $$k(6k+1)=1+q+q^2+\cdots+q^n$$ with $q$ a prime power?
2) For which positive ...
- 23k
2
votes
2
answers
320
views
A graphic representation of classical unitals on 28 points
I would like to understand the geometry of the classical unitals.
They are block designs containing $q^3+1$ points and whose blocks have cardinality $q+1$, where $q$ is a prime power. For $q=2$ (if I ...
- 41.8k
2
votes
3
answers
319
views
Constructions of $2-(v,3,3)$-designs
I am looking for ways to construct an infinite family of designs with parameters $2-(v,3,3)$ and apart from some doubling-type recursive constructions (such as in this paper) I haven't found anything ...
- 6,962
0
votes
1
answer
157
views
"JigSaw Puzzle" on Set Family
One of my research problem can be reduced to a question of the following form
Given a set family $\mathcal{F}$ of $[n]$ , such that every element of $[n]$ lies in exactly $K$ sets in $\mathcal{F}$, ...
- 384