All Questions
Tagged with design-theory or combinatorial-designs
125 questions
4
votes
2
answers
620
views
Is Ryser's conjecture on permanent minimizers still open?
Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$.
Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that if $A(k,n)$ ...
- 6,962
4
votes
2
answers
221
views
Is the domination number of a combinatorial design determined by the design parameters?
Let $D$ be a $(v,k,\lambda)$-design. By the domination number of $D$ I mean the domination number $\gamma(L(D))$ of the bipartite incidence graph of $D$.
Is $\gamma(L(D))$ determined only by $v,k$, ...
- 6,962
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
"Codes" in which a group of words are pairwise different at a certain position
I read the following problem, claimed to be in the IMO shortlist in 1988:
A test consists of four multiple choice problems, each with three options, and the students should give an unique answer to ...
- 2,581
4
votes
2
answers
331
views
Is there an infinite number of combinatorial designs with $r=\lambda^{2}$
A quick look at Ed Spence's page reveals two such examples: (7,3,3) and (16,6,3).
If there is a known classification and/or name by which such designs go, I'd love to know about them too.
EDIT: I ...
- 6,962
4
votes
1
answer
170
views
About $CW(512,16^2)$
Definitions: A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{ij}$ in $\{0, \pm 1\}$ such that $WW^T=kI$,
where $I$ is the identity matrix. A circulant ...
- 696
4
votes
3
answers
781
views
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?
It seems to me that the answer should be yes, but my naive attempts to come up with an example have failed.
Just to clarify, by finite hyperbolic geometry I mean a finite set of points and lines such ...
- 1,701
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
4
votes
1
answer
113
views
Bounding the number of orthogonal Latin squares from above
As is usual, let $N(n)$ denote the maximum size of a set of mutually orthogonal Latin squares of order $n$. I am wondering what results hold that bound $N(n)$ from above; the only ones I can think of ...
- 513
4
votes
2
answers
262
views
Known result about existence of $n$-vertex $k$-uniform $r$-hypergraphs?
Are there known results about $n,k,r$ such that $n$-vertex $k$-uniform $r$-regular hypergraphs exist? If this is too large a class of hypergraphs, what if $k=\tilde{\theta}(\sqrt{n})$? What if an ...
- 281
4
votes
1
answer
81
views
k-partite design
Is the following true? For every $n \geq 1, k\geq 2$, there is a set $S \subseteq [n]^k$ of size $|S| = n^2$ such that every two $k$-tuples in $S$ have at most one common entry.
Does anyone know if ...
- 428
4
votes
1
answer
70
views
Balancing out edge multiplicites in a graph
Let $G$ be a multigraph with maximum edge multiplicity $t$ and minimum edge multiplicity $1$ (so that there is at least one 'ordinary' edge).
Is there some simple graph $H$ such that the $t$-fold ...
- 1,081
4
votes
0
answers
89
views
Software reference for combinatorial design
If one were to require quick and easy access to sizeable latin squares, room squares, Steiner systems, designs, balanced block designs... where to look, what software to use?
- 1,461
4
votes
0
answers
77
views
Existence of finite 3-dimensional hyperbolic balanced geometry
Together with @TarasBanakh we faced the problem described in the title. Let me start with definitions.
A linear space is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of ...
- 501
4
votes
0
answers
350
views
Existence of a block design
Let $\ell$ be an integer parameter. I want to ask the existence of the following design: There is a universal constant $\beta < 1$ such that for all sufficiently large $\ell$, the following holds:
...
- 143
4
votes
0
answers
171
views
Reduction argument from a general vertex set V(G) to a prime power in Prof. Keevash's proof on the Existence of Designs
The proof flow of the paper "On the Existence of Designs" by Prof. Keevash as I understand it is the following:
-- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3)
-- Covering ...
- 81
4
votes
0
answers
163
views
Number of cyclic difference sets
A subset $D=\{a_1,\ldots,a_k\}$ of $\mathbb{Z}/v\mathbb{Z}$ is said to be a $(v,k,\lambda)$-cycic difference set if for each nonzero $b\in\mathbb{Z}/v\mathbb{Z}$, there are exactly $\lambda$ ordered ...
- 767
3
votes
2
answers
304
views
When do such regular set systems exist?
Let '$n$-set' mean 'a set with $n$ elements'.
May we choose $77=\frac16\binom{11}5$ 5-subsets of 11-set $M$ such that any 6-subset $A\subset M$ contains unique chosen subset? Positive answer to ...
- 109k
3
votes
2
answers
1k
views
How many elements with a hamming distance of 3 or less?
[This is a complete rewrite which makes some of the comments redundant or irrelevant.]
Take a set of $50$ elements. How many subsets of size $5$ are needed so that every subset of size $5$ will ...
- 33
3
votes
1
answer
294
views
Smallest number of subsets whose squares cover the whole square
Let $2 \leq k \leq n$ be integers, let $[n] := \{1,2,\ldots,n\}$, and for a subset $A \subseteq [n]$ let $A^2 := A \times A$ be the Cartesian product of $A$ with itself and let $|A|$ denote the ...
- 6,351
3
votes
1
answer
153
views
Cycling through a general combinatorial design on $\omega$
This is a generalisation of an older question inspired by a football tournament (which does not have an answer yet).
Let $\frak P$ be a partition of $\omega$ into blocks, that is, pairwise disjoint ...
- 48.9k
3
votes
1
answer
427
views
Ranks of higher incidence matrices of designs
In 1978 Doyen, Hubaut and Vandensavel proved that if $S$ is a Steiner triple system $S(2,3,v)$ then the $GF(2)$ rank of its incidence matrix $N$ is
$$
Rk_{2}(N)=v-(d_{p}+1),
$$
where $d_{p}$ is the ...
- 6,962
3
votes
3
answers
707
views
Orthogonal Latin Square 6*6
I need to make remarks about Tarry's Proof for the nonexistence of 6x6 Latin Squares as part of my final exam for a class I'm in. Problem is, I can't find it ANYWHERE on the internet. I can only find ...
- 39
3
votes
1
answer
150
views
Mutually orthogonal Latin hypercubes
A $d$-dimensional Latin hypercube with side length $n$ is a $d$-dimensional array with $n$ symbols such that along any line parallel to an axis, each symbol appears exactly once. Let us call a $(n,d)$ ...
- 675
3
votes
1
answer
83
views
For which sets of $(n, m, k)$ does there exist an edge-labelling (using $k$ labels) on $K_n$, such that every single-labelled subgraph is $K_m$?
Or, equivalently - for what sets of $(n, m, k)$ is it possible, for a group* of $n$ people, to arrange $k$ days of "meetings", such that every day the group is split into subgroups of $m$ people, and ...
- 131
3
votes
1
answer
1k
views
Residual design (BIBD) with repeated blocks
Simple BIBD are defined as those designs in which incindence relation is "is element". So effectively blocks are subsets of points. Equivalently there should be no "repeating blocks" ie. blocks that ...
- 95
3
votes
1
answer
120
views
On the existence of a certain graph/hypergraph pair
Let $V$ be a finite set, $G$ a simple graph with vertex set $V$, and $H$ a hypergraph (i.e., set of subsets) with vertex set $V$ satisfying the following three conditions:
each pair of elements of $V$...
- 24.2k
3
votes
1
answer
222
views
Has this kind of design been studied before?
Consider a design $(X,\mathcal{B})$, satisfying:
Each block in $\mathcal{B}$ has the same size
The intersection of every two blocks has the same size
Of course, it is easy to find many examples of ...
- 63
3
votes
1
answer
275
views
Lower bounding the maximum size of sets in a set family with union promise
The following problem has come up while working on the relationship between certificate and randomized decision tree complexities of boolean functions. However, I think it is of interest by itself and ...
3
votes
0
answers
54
views
Cliques in Incomplete block designs
I'm interested in inequalities that guarantee the presence of cliques in incomplete block designs. Here's the set-up:
I have an incidence structure $(V, B)$ which is an incomplete block design: $V$ is ...
- 11.2k
3
votes
0
answers
148
views
Linear combinations of special matrices
I am a hobby computer scientist and I have a problem to which I am searching an efficient algorithm.
Given an integer n, we want to combine some square input-matrices of size n in a way that is ...
- 73
3
votes
0
answers
130
views
Graeco-Latin squares and outer-automorphisms
It is well known that $n=6$ is the only number greater than two in which there is no Graeco-Latin square of order $n$. It is also well known that $n=6$ is the only number greater than two in which ...
- 2,649
3
votes
0
answers
92
views
what is the largest real orthogonal design in $n$ variables?
A real orthogonal design in $n$ variables is an $m \times n$ matrix with
entries from the set $\pm x_1,\pm x_2,\cdots,\pm x_n$ that satisfies :
$$ A A^T = (x_1^2 + x_2^2 + \cdots x_n^2) I_m $$
...
- 451
3
votes
0
answers
129
views
A question on the behavior of intersections of certain block design
Let $[d]$ be a universe and $S_1, \dots, S_m$ be an $(\ell, a)$-design over $[d]$ which means that:
$\forall i \in [m], S_i \subseteq [d], |S_i|=\ell$.
$\forall i \neq j \in [m]$, $|S_i \cap S_j| \...
- 143
3
votes
0
answers
317
views
Prime Hadamard matrices
Assume that $n$ is a sufficiently large number. Is there a Hadamard matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, k}=1$)...
- 321
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
2
votes
2
answers
298
views
Resolvable designs from projective space
Resolvable designs are block designs with the additional property that the blocks can be partitioned into partitions of the points. It is easy to see that lines in affine space form a resolvable ...
- 7,633
2
votes
2
answers
802
views
Minimally intersecting subsets of fixed size
The question I have, is how to generate the following collection of subsets:
Given a set $S$ of size $n$. I want to find a sequence of $k$ subsets of fixed size $m$, $0<m<n$, such that at each ...
- 145
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
1
answer
60
views
Constructing Group Divisible Designs - Algorithms?
I am starting my research on group divisible designs this year and I wonder if there are any algorithms/software that help with constructions.
Thank you
- 29
2
votes
1
answer
98
views
One question about nega-cyclic Hadamard matrices
Let $n$ be a multiple of $4$, is there any $n \times n$ negacyclic Hadamard matrix? If yes - how to construct it? If no - why?
Here an $n \times n$ nega-cyclic matrix is a square matrix of the form:
\...
- 696
2
votes
1
answer
100
views
Relative difference sets?
Do you know how to find order of known groups with RDS or without? Or Known groups which have RDS or not? If there is a references to survey all researches?
user91523
2
votes
2
answers
217
views
Minimal number of blocks in a $(n,n/2,\lambda)$ block design
A $(n,n/2,\lambda)$ block-design is a family $A_1,...,A_K$ of subsets of $[n]$ such that $|A_i|=n/2$ and for every $1 \leq i < j \leq n$ it holds that
$\#\{1 \leq k \leq K : i,j \in A_k \} = \...
- 21
2
votes
1
answer
221
views
Hitting sets (aka covers aka transversals) of Steiner triple systems
Does there exist a constant $c$ so that the lines of every Steiner
triple system on $v$ points can be covered by $cv$ points?
That is if $D \in STS(v)$ with point set $T=\{1,2,\ldots,v\}$ then ...
- 6,962
2
votes
1
answer
130
views
Distinguishing points by sets of given size
The problem is:
Given a finite set $X$ with size $x$ and let $B$ denote a family of $k$-element subsets of $X$, called blocks. What is the smallest possible number $n$ of blocks such that every ...
- 9,501
2
votes
4
answers
313
views
What is the correct term for "co-covering" designs
An (n, k, l) covering design is a family of k-subsets of an n-element set such that every l-subset is contained in at least one of them. Now, what is the correct term for a family of k-subsets such ...
2
votes
1
answer
169
views
Lower bounds on cardinality of a union of blocks in a design
Let $D$ be a $(v,k,\lambda)$-design (repeated blocks are allowed). I would like to get a lower bound on the cardinality of the union of $s$ blocks. A naive application of inclusion-exclusion gives $sk-...
- 6,962
2
votes
1
answer
327
views
Existence of Steiner system designs given $n,k,t$
I am familiar with the recent Keevash paper here which proves that given some $t,n,k,\lambda$ then provided standard divisibility conditions hold, and $n$ is suitably large, there exists a $t-(n,k,\...
2
votes
1
answer
427
views
Popular elements in cross-intersecting families
Let $\mathcal{T}$ and $\mathcal{S}$ be two families of subsets of $[n]$ such that for all $T_i\in \mathcal{T}$ and $S_j\in \mathcal{S}$,
$|T_i \cap S_j| \neq\emptyset$
$|T_i| , |S_j| \leq t = O(\log(...
- 21
2
votes
0
answers
187
views
Matrix with elementary symmetric polynomials as entries
Let $n\geq 1$, and for each $j=1,\ldots, n+1$ let $\mathbf{X}_{j}=(X_{j1},\ldots, X_{jn})$ be $n$ variables. Let $M$ be the $(n+1)\times (n+1)$ matrix whose $(i,j)$-th entry is $$M_{ij}=(-1)^i e_{i-1}(...
