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Given a set of $v$ numbers. Fix $v$. How many sets of $v$ sorted triples can be created, matching the following conditions:

  • two triples shall have atmost one number in common
  • over all triples each number shall appear three times?

A representative example set of triples with $v=8$: $$ S_A=(\color{red}1,\color{blue}2,\color{green}5)\;;\;S_B=(\color{blue}2,\color{grey}3,\color{purple}6)\;;\;S_C=(\color{grey}3,\color{pink}4,\color{orange}7)\;;\;S_D=(\color{red}1,\color{pink}4,8) $$ $$ S_E=(\color{blue}2,\color{orange}7,8)\;;\;S_F=(\color{grey}3,\color{green}5,8)\;;\;S_G=(\color{pink}4,\color{green}5,\color{purple}6)\;;\;S_H=(\color{red}1,\color{purple}6,\color{orange}7) $$

Can this be seen as $(v, b, r, k, λ)$ block designs? When I read the definition there I find:

  • $v(=8)$: points, number of elements of X
  • $b(=8)$: number of blocks
  • $r(=3)$: number of blocks containing a given point
  • $k(=3)$: number of points in a block
  • $\lambda(=1)$: number of blocks containing any 2 distinct points

But obviously this doesn't fulfill the condition ${\displaystyle \lambda (v-1)=r(k-1)}$...

UPDATE My example looks like a subset of elements of the Steiner Triple System $STS(9)$, where subsets that contain $9$ are omitted.

The question is motivated by bipartite cubic graphs. Bipartiteness forces $v=b$, cubicity asks for $r=k=3$. $\lambda =1 $ means that the graph is simple and has no squares.

The example above is derived from the graph shown here. It has 6 octagons living on a double torus. Black points are labelled with numbers $1$ to $6$; white points are labelled with characters $A$ to $H$.

draks ...
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    Your definition is not equivalent to having a $\lambda=1$ in the design sense. Having $\lambda = 1$ would mean every pair of points would be contained in a unique block, in your example you have every pair of points contained in at most one block. I think this would sometimes be referred to as a partial design, or a $(v,v,3,3,{0,1})$ partial design. – xxxxxxxxx Jan 11 '19 at 17:05
  • @MorganRodgers partial design, I'll check that, thanks ... – draks ... Jan 12 '19 at 12:48
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    The requirement that every pair (of triplets) contains at most 1 common number actually reminds me of codes, although the connection is loose enough that I'm not sure it helps.. A set meeting your requirement can be visualized as a $v \times v$ binary matrix $A$, where each row is a number and each column is a triplet, and $A_{ij} = 1$ iff number $i$ is in triplet $j$. By your requirements, every row and every column of this matrix has exactly three $1$s. Now you can consider the rows as codewords. – antkam Jan 16 '19 at 19:13
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    (cont'd) The interesting thing is that these codewords are at least distance-4 apart, making them Single-Error-Correcting, Double-Error-Detecting (SECDED). Such codes are in wide use, see e.g. https://en.wikipedia.org/wiki/Hamming_code Hamming codes have a lot of symmetries, but I've never seen these studied in the context of restricting the codes words to each having exactly three bits being $1$. So this connection may indeed be too loose to be useful. – antkam Jan 16 '19 at 19:17
  • @antkam interesting any way. Thanks. If you can find a reference to SECDED or working out the analogy: Worth an answer... – draks ... Nov 28 '19 at 18:17

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