# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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. ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### “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 ...
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 ...