Questions tagged [combinatorial-designs]

Design theory is the subfield of combinatorics concerning the existence and construction of highly symmetric arrangements. Finite projective planes, latin squares, and Steiner triple systems are examples of designs.

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11 votes
1 answer
372 views

Does every finite affine plane have the doubling property?

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
5 votes
5 answers
545 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 ...
4 votes
0 answers
58 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 ...
2 votes
2 answers
303 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 ...
15 votes
1 answer
866 views

Scrambling a “Connections” grid

Given a 4-by-4 array of distinct words, is it possible to scramble the array in four different ways in such a fashion that each possible word-pair appears adjacently in one of the five arrays (the ...
3 votes
1 answer
285 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 ...
4 votes
0 answers
75 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?
7 votes
1 answer
1k views

Are there infinite constructions for partial circulant Hadamard matrices?

I believe that the circulant Hadamard conjecture (that there are no circulant Hadamard matrices of size greater than $4\times4$) is still open. I also know that examples of $(n/2) \times n$ matrices ...
55 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, ...
0 votes
0 answers
40 views

Minimal m such that m x K_n is decomposable into disjoint C_3

For a given $n$, is there a way to calculate the minimal value $m$ such that you can decompose the multigraph: $$m \times K_n$$ into disjoint 3-cycles? What about a more general result applied to ...
11 votes
2 answers
805 views

On the Steiner system $S(4,5,11)$

Is there a nice way to partition the edges of the complete $5$-uniform hypergraph on $11$ vertices into $7$ copies of the Steiner system $S(4,5,11)$? If this is obvious or elementary, I apologize in ...
1 vote
1 answer
72 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: \...
3 votes
1 answer
127 views

On the half-skew-centrosymmetric Hadamard matrices

Definition 1: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal. Definition 2: A matrix $A$ is half-skew-centrosymmetric if ...
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 $\mathcal{F}$, ...
0 votes
0 answers
81 views

Bounds for smallest non-trivial designs

Given $s>t\ge 2$, let $N(s,t)$ be the smallest integer $n>s$ such that there exists an “$(n;s;t;1)$-design” (i.e., a collection of $s$-subsets $e_1,\dots,e_m$ of $[n]:=\{1,\dots,n\}$, such that ...
6 votes
1 answer
143 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 ...
1 vote
1 answer
131 views

On the existence of symmetric matrices with prescribed number of 1's on each row

We are considering the following problem: Given an integer $n$ and a sequence of integers $r_i,\ 1\le i\le n$, with $0\le r_i\le n-1$ does there exists a symmetric matrix $A$ such that the diagonal ...
9 votes
1 answer
303 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 ...
6 votes
0 answers
3k views

A generalization of covering designs and lottery wheels

This question is inspired by a recent problem . A $(v,k,t)$ covering design is a pair $(V,B)$ where $V$ is a set of $v$ points and $B$ is a family of $k$ point subsets (called blocks) such that ...
11 votes
2 answers
656 views

$\mathbb Z/p\mathbb Z=A\cup(A-A)$?

$\newcommand{\Z}{\mathbb Z/p\mathbb Z}$ Can one partition a group of prime order as $A\cup(A-A)$ where $A$ is a subset of the group, $A-A$ is the set of all differences $a'-a''$ with $a',a''\in A$, ...
3 votes
1 answer
149 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 ...
1 vote
0 answers
101 views

On a combinatorial design inspired by a football (soccer) tournament

Real-world inspiration. My younger son was playing a micro football (soccer) tournament this afternoon with $3$ other friends. Let's label the $4$ kids $0,1,2,3$. They played $3$ matches: $\{0,1\} \...
5 votes
0 answers
885 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 ...
2 votes
2 answers
212 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 \} = \...
1 vote
1 answer
136 views

3-partition of a special set

$S_5$ is a set consisting of the following 5-length sequences $s$: (1) each digit of $s$ is $a$, $b$, or $c$; (2) $s$ has and only has one digit that is $c$. $T_5$ is a set consisting of the following ...
4 votes
3 answers
707 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 ...
8 votes
3 answers
416 views

Latin squares with one cycle type?

Cross posting from MSE, where this question received no answers. The following Latin square $$\begin{bmatrix} 1&2&3&4&5&6&7&8\\ 2&1&4&5&6&7&8&3\\...
4 votes
1 answer
79 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 ...
1 vote
0 answers
55 views

Are sharper lower bounds known for these potentials on the sphere?

Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that $$ \sum_{i=1}^n\sum_{j=1}^n(x_i\cdot x_j)^{2\ell}\geq\frac{(2\ell-1)!!(d-2)!!}{(d+2\ell-2)!!}\cdot n^2, $$ with ...
16 votes
3 answers
1k 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 ...
3 votes
0 answers
135 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 ...
3 votes
1 answer
137 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 $(n,d)$ ...
3 votes
0 answers
52 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 ...
1 vote
0 answers
51 views

Optimal choice of points to maximize majorities in a $t-(v,k,\lambda)$ design

Let us consider a design $\mathcal{D} = (V,\mathcal{B})$ with points in $V$ and blocks in $\mathcal{B}$. I am interested in the special case of a $t-(v,k,\lambda)$ design for $k=3$, i.e., all blocks ...
11 votes
5 answers
492 views

What are efficient pooling designs for RT-PCR tests?

I realize this is long, but hopefully I think it may be worth the reading for people interested in combinatorics and it might prove important to Covid-19 testing. Slightly reduced in edit. The ...
12 votes
1 answer
193 views

Ternary sequences satisfying $ x_i + y_i = 1 $ for some $ i $

Consider a set of strings $ {\mathcal S} \subset \{0, 1, 2\}^n $ satisfying the following two conditions: 1.) every string in $ {\mathcal S} $ has exactly $ k $ symbols from $ \{0, 1\} $ (i.e., $ \...
3 votes
1 answer
72 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 ...
1 vote
1 answer
66 views

Does there exist a non-degenerate symmetric combinatorial 3-design?

Is there a non-degenerate 3-design where the number of blocks equals the number of points? Non-degenerate in this context means that a point is incident with at least 2 and at most #blocks-2 blocks.
2 votes
0 answers
83 views

Packings with block size equal to $6$?

In design theory the following is the defintion of a packing : Definition : A $(v,k)$-packing is a pair $(V, \mathcal{B})$ of a finite set $V$ of cardinality $\vert V \vert = v$ and a finite set $\...
4 votes
3 answers
235 views

Best strategy for a combinatorial game

Consider the following scenario. We have 20 balls and 100 boxes. We put all 20 balls into the boxes, and each box can contain at most one ball. Now suppose we are given 5 chances to pick 20 out of ...
5 votes
2 answers
195 views

Coloring in Combinatorial Design Generalizing Latin Square

I have a question about a combinatorial design very similar to a Latin Square, which is arising out of an open problem in graph theory. The design is an $n \times n$ matrix whose entries we want to ...
8 votes
2 answers
555 views

Pfaffian representation of the Fermat quintic

It is known (see for instance Beauville - Determinantal hypersurfaces) that a generic homogeneous polynomial in $5$ variables of degree $5$ with complex coefficients can be written as the Pfaffian of ...
3 votes
0 answers
129 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 ...
10 votes
2 answers
357 views

Lower bound for a combinatorial problem ($N$ students taking $n$ exams)

We have $N$ students and $n$ exams. We need to select $n$ out of the students using the grade of those exams. The procedure is as follows: 1- We set some ordering on the exams. 2- Going through this ...
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
4 votes
1 answer
90 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 ...
10 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?
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 $V$...
2 votes
1 answer
129 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 ...
4 votes
1 answer
69 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 ...