The combinatorial-designs tag has no wiki summary.

**6**

votes

**1**answer

148 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 ...

**3**

votes

**1**answer

447 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 ...

**10**

votes

**1**answer

215 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.
...

**4**

votes

**2**answers

343 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 ...

**2**

votes

**1**answer

101 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 ...

**1**

vote

**0**answers

53 views

### Point sets with tangents through every point

Let $D=(P,L)$ be either a $(v,k,\lambda)$-design or a near-linear space (or, more generally, any incidence structure with "points" and sets of points which are called "blocks" or "lines") and let $S ...

**3**

votes

**3**answers

129 views

### Isomorphism testing in STS(13)

What is the simplest isomorphism invariant which can distinguish between the two non-isomorphic Steiner triple systems on $13$ points?
Train structure and cycle structure, as described here, do the ...

**3**

votes

**2**answers

151 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 ...

**8**

votes

**3**answers

346 views

### colorings of ${\mathbb Z}^d$ with constraints

For a lattice $\mathbb Z^d$, denote by lattice line any line that contains two (and thus infinitely many) lattice points.
For $2\le k<n$, define a $(n,k)$-coloring, or $C_d(n,k)$ for short, ...

**1**

vote

**1**answer

252 views

### 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 ...

**0**

votes

**1**answer

175 views

### a block design question

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 ...

**2**

votes

**1**answer

131 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 ...

**1**

vote

**3**answers

199 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 ...

**7**

votes

**0**answers

346 views

### Does $(\mathbb{Z}/n\mathbb{Z})^2$ ever admit a difference set when $n$ is odd?

A difference set of a group $G$ is a subset $D\subseteq G$ with the property that there exists an integer $\lambda>0$ such that for every non-identity member $g$ of $G$, there exist exactly ...

**14**

votes

**1**answer

576 views

### Octonions and the Fano plane.

Does the Fano plane mnemonic for octonion multiplication have any deeper meaning?
http://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg
The symmetry group of the Fano plane is PSL(2,7), ...

**7**

votes

**0**answers

162 views

### Tenacious structure

Let $\def\A{\mathbb A}\def\F{\mathbb F}\F_3$ be the Galois field with three elements and let $\A^d=\A^d(\F_3)$ be the affine space of dimension $d$ over $\mathbb F_3$ â€”the subject is combinatorics and ...

**2**

votes

**1**answer

77 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 ...

**3**

votes

**2**answers

339 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 $A(k,n)$ ...

**2**

votes

**2**answers

141 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 ...

**3**

votes

**3**answers

364 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 ...

**1**

vote

**5**answers

263 views

### Pairwise balanced designs with $r=\lambda^{2}$

A while ago I asked how to construct an infinite family of $(v,b,r,k,\lambda)$-designs satisfying $r=\lambda^{2}$ and got very good answers from Yuichiro Fujiwara and Ken W. Smith.
Now I'd like to up ...

**4**

votes

**2**answers

248 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 ...

**5**

votes

**1**answer

221 views

### Block design question

Given fixed values for $d \leq k \leq v$. I would like to find a set $B$ of $d$-sets of $[v]$ with the following properties:
Every $k$-set of $[v]$ contains at least one element of $B$
Every ...

**11**

votes

**5**answers

461 views

### Intersecting 4-sets

Is it possible to have more than $N = \binom{\lfloor n/2\rfloor}{2}$ subsets of an $n$-set, each of size 4, such that each two of them intersect in 0 or 2 elements?
To see that $N$ is achievable, ...

**2**

votes

**1**answer

378 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 = ...

**-1**

votes

**1**answer

298 views

### 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: ...

**3**

votes

**1**answer

331 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 ...

**6**

votes

**3**answers

579 views

### Meeting management

A friend wants to have ten meetings of six people every day for five days with no pair of people meeting twice. Is this possible? It appears to be a question about maximal decomposition of a complete ...