Questions tagged [steiner-triple-system]

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Existence of finite 3-dimensional hyperbolic balanced geometry

Together with @TarasBanakh we faced the problem described in the title. Let me start with definitions. A linear space is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of ...
Ihromant's user avatar
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1 vote
1 answer
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Steiner-like systems on $\omega$

Let $S(\omega)$ denote the collection of "sparse" infinite subsets of $\omega$, that is, $X\subseteq \omega$ is a member of $S(\omega)$ if and only if both $X$ and $\omega\setminus X$ are ...
Dominic van der Zypen's user avatar
4 votes
1 answer
322 views

Enumerating subsets with no triple appearing together more than once

This question is motivated by a real-world application related to an art project that involves displaying images, but my search hit a dead end after finding the wikipage about Kirkman systems (other ...
Benjamin Dickman's user avatar
1 vote
0 answers
51 views

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 ...
mgus's user avatar
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2 votes
1 answer
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List coloring of a graph corresponding to a Steiner triple system

Consider the Steiner triple system $S(2,3,n)$ for a suitable integer $n$. We define a graph $G$ with all the vertices as precisely the blocks of the above steiner triple system and any two points ...
vidyarthi's user avatar
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1 vote
1 answer
212 views

From Steiner systems to geometric lattices to matroids

I am looking for a specific matroid. I found a source that claimed to discuss these matroids, but then, only discusses geometric lattice. Even more, in that paper, the geometric lattice that seems to ...
M. Winter's user avatar
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9 votes
0 answers
179 views

Sections of "forgetful" projections between flag manifolds

Given a subset $S\subseteq\{1,\cdots,n\}$ there is an associated flag manifold $F(S)$. Whenever $A\subseteq B$ there is a "forgetful" projection $F(A)\leftarrow F(B)$ (in fact I think its fibers are ...
anon's user avatar
  • 441
4 votes
2 answers
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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 ...
Gorka's user avatar
  • 1,825
2 votes
1 answer
207 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 ...
Felix Goldberg's user avatar
6 votes
3 answers
448 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 ...
Felix Goldberg's user avatar
-1 votes
1 answer
377 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: $\...
Cindy's user avatar
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3 votes
2 answers
433 views

Can a partial Steiner triple system be completed?

This is probably well-known... but I am afraid the literature on this subject bewilders me a little bit: Suppose we have a partial Steiner triple system, whereby I mean a finite set $E$ and a set $...
Mariano Suárez-Álvarez's user avatar
11 votes
4 answers
5k views

Constructing Steiner Triple Systems Algorithmically

I want to create STS(n) algorithmically. I know there are STS(n)s for $n \cong 1,3 \mod 6$. But it is difficult to actually construct the triples. For STS(7) it is pretty easy and but for larger n I ...
JarvisP's user avatar
  • 133
1 vote
1 answer
483 views

Why is a block graph of a Steiner Triple System is a Strongly Regular Graph?

With parameters: srg(v(v-1)/6, 3(v-3)/2, (v-3)/2, 9) Should be straightforward counting which alludes me... Thanks! Shay
Shay's user avatar
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4 votes
1 answer
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Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and $q$-binomials?

The short version of my question is: 1)For which positive integers $k, n$ is there a solution to the equation $$k(6k+1)=1+q+q^2+\cdots+q^n$$ with $q$ a prime power? 2) For which positive ...
Daniel Litt's user avatar
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13 votes
1 answer
2k views

Database of Steiner triple systems

Can anyone point me to an online database of Steiner triple systems? My Google-fu is only getting me to descriptions of the few smallest ones, mostly Google book scans (which are rather useless to ...
Mariano Suárez-Álvarez's user avatar
4 votes
1 answer
534 views

Diagonally-cyclic Steiner Latin squares

A Steiner triple system is a decomposition of $K_n$ into $K_3$, such as $S=\{013,026,045,124,156,235,346\}$. Steiner triple systems give rise to a Steiner Latin squares, such as $L$ below. \[L=\left(...
Douglas S. Stones's user avatar