# Tagged Questions

The tag has no usage guidance.

268 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: ...
98 views

173 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 ...
189 views

An $m\times n$ matrix with entries $\pm 1$ is said to be partial Hadamard if any two rows are orthogonal. See Reference for partial Hadamard matrices. Given $n\equiv 0\,(\mathrm{mod}\,4)$, what is the ...
43 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 ...
175 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 ...
368 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, ...
389 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 ...
190 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 ...
181 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 ...
224 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 ...
485 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 ...
604 views

### What is the largest number of k-element subsets of a given n-element set S such that…

Given a set S of n elements. What is the largest number of k-element subsets of S such that every pair of these subsets has at most one common element?
716 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), ...
429 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 ...
169 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 $\mathbb F_3$ —the subject is combinatorics and ...
89 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 ...
406 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 $A(k,n)$ ...
161 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 ...
415 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 ...
288 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 ...
2k views

### What are the major open problems in design theory nowaday?

I gather that the question whether the Bruck-Chowla-Ryser condition was sufficient used to top the list, but now that that's settled - what is considered the most interesting open question?
276 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 ...
249 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 ...
481 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, ...