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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48 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 ...
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1answer
44 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.
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Existence of specific 2-designs (BIBDs)

I am looking for constructions of 2-(n,b,r,k,2)-designs of the following type, given an integer $t$ with $t=\max(r,k)$: -every point is in $r$ blocks -every block has size $k$ -there are $n=\Theta(t^...
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76 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 $\...
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203 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 ...
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2answers
133 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 ...
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110 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 ...
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316 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 ...
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1answer
50 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
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1answer
57 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 ...
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1answer
106 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 $V$...
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1answer
118 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 ...
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448 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 ...
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2answers
135 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 ...
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189 views

Minimum covers of complete graphs by $4$-cycles

I am interested in coverings of the (edge set of the) complete graph $K_n$ by cycles of length $4$. It is clear that such coverings exist for each $n \ge 4$. I need to find the minimum number of $4$-...
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1answer
79 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?
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1answer
190 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 ...
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A generalization of PBIBDs

A PBIBD is an incidence structure together with an underlying symmetric association scheme. One can relax the symmetry requirement, and ask for incidence structures with an underlying (not necessarily ...
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88 views

Steiner-like systems with large edges and many intersections

Let $l\geq 3$ be an integer. Is there $n\in\mathbb{N}$ and a hypergraph $H=(\{1,\ldots,n\},E)$ with the following properties? for all $e\in E$ we have $|e| \geq l$ $e_1\neq e_2 \in E \implies |e_1 \...
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3answers
373 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 ...
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177 views

Non-uniform Ray-Chaudhuri-Wilson (generalized Fisher's inequality)

A $t$-design on $v$ points with block size and index $\lambda$ is a collection $\mathcal{B}$ of subsets of a set $V$ with $v$ elements satisfying the following properties: (a) every $B\in\mathcal{B}$ ...
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86 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 : $$ A A^T = (x_1^2 + x_2^2 + \cdots x_n^2) I_m $$ ...
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65 views

Non-continuous behaviour when designing a repeated experiment

Assume one can perform measurements of an unknown quantity $\theta$ as $$y = \theta + \epsilon(t),$$ where $\epsilon(t) \sim \mathcal{N}(0,1/t)$ is the measurement error when a time $t$ was spent to ...
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2answers
138 views

Reference Request: “Resolutions” of $K_n$ for $n$ odd

A resolution (in the combinatorial design sense) of $K_{n}$ is a collection of sets of edges of $K_{n}$ so that within each set of edges, each vertex appears once, and over the entire collection, each ...
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577 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 $...
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174 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 ...
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2answers
172 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 \} = \...
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2answers
232 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 $...
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1answer
215 views

Covering designs where $v$ is linear in $k$

A $(v,k,t)$ covering design is a collection of $k$-subsets of $V=\{1,\ldots,v\}$ chosen so that any $t$-subset of $V$ is contained in (or "covered by") at least one $k$-set in the collection. ...
3
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295 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 ...
5
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1answer
350 views

reverse definition for magic square

Recently, I saw a question in see here which is so interesting for me. This question is as follows: Is it possible to fill the $121$ entries in an $11×11$ square with the values $0,+1,−1$, so that ...
6
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1answer
278 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 ...
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1answer
41 views

Vector version of balanced incomplete block designs

I am interested in finding out what is known about the following generalization of balanced incomplete block designs (BIBDs): "What is the maximum size of a collection $B$ of $v$-dimensional unit ...
4
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330 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: ...
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117 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| \...
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4answers
227 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 ...
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1answer
268 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,\...
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693 views

Combinatorial designs textbook recommendation

Good evening, I am currently taking a class which has combinatorial designs as the first topic, we are using Peter Cameron's book Designs, Graphs, Codes and their Links which I am finding extremely ...
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483 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 ...
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2answers
440 views

covering designs of the form $(v,k,2)$

A covering design $(v,k,t)$ is a family of subsets of $[v]$ each having $k$ elements such that given any subset of $[v]$ of $t$ elements it is a subset of one of the sets of the family. A problem is ...
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2answers
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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} $ ...
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167 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 ...
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1answer
700 views

What measurable quantity can constrain the number of odors human can discriminate?

This is not a very typical MO question, but I hope you bear with me. It concerns a recent disagreement in the biology literature about how many different odors humans can discriminate. The authors of ...
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1answer
109 views

All $2$-designs arising from the action of the affine linear group on the field of prime order

Let $p$ be a prime and $\mathbb{Z}_p$ denote as usual the field of order $p$. Let $AL(p)$ be the affine linear group $\{x\mapsto ax+b \;|\; a\in \mathbb{Z}_p\setminus \{0\}, b\in\mathbb{Z}_p\}$. For a ...
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130 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 ...
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1answer
131 views

Number of points in an intersecting linear hypergraph

I first asked the question below at math.stackexchange.com ( https://math.stackexchange.com/questions/920442/number-of-points-in-an-intersecting-linear-hypergraph ) but somebody suggested I ask it in ...
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1answer
329 views

Is there a simple proof that there is no five mutually orthogonal Latin squares of order 6?

It is well known that there is a projective plane of order $n$ if and only if there exist a set of $n-1$ mutually orthogonal Latin squares. The first nontrivial case is $n=6$, which fails because of ...
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1answer
953 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 ...
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424 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 advance.
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409 views

Can we sometimes define the parity of a set?

Suppose that ${n\choose k}, {n-1\choose k-1}, \ldots, {n-k+1\choose 1}$ are all even. (This happens for example if $k=2^\alpha-1$ and $n=2k$.) In this case, can we select ${n\choose k}/2$ sets of size ...