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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Known results on cyclic difference sets
Is there any infinite family of $v$ for which all the $(v,k,\lambda)$-cyclic difference sets with $k-\lambda$ a prime power coprime to $v$ have been determined?
A subset $D=\{a_1,\ldots,a_k\}$ of $\...
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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:
\...
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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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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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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 ...
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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\} \...
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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 ...
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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 ...
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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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"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 $\mathcal{F}$, ...
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a block design question: Does every special 1-design admit a partition which respects enough of the blocks?
Is it possible to show that every 1-design $D$ with $\lambda=4,k=4$ on $v$ points (for $v$ that is a multiple of $3$) contain some 1-design $Q$ with $\lambda=1,k=3$ on $v$ points such that every block ...
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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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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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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 ...
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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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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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Number of blocks in a t-(v,k,l) design with empty intersection with a given set U [closed]
Question
Given a $t-(v,k,\lambda)$ design $(X,\mathcal{B})$ and a set $U\subset X$ with $|U|=u\leq t$, what is the number of blocks $B\in\mathcal{B}$ such that $B\cap U=\emptyset$?
The answer is: $\...
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