id: 06093198
dt: j
an: 06093198
au: Pettis, C.S.
ti: The triangle intersection problem for hexagon triple systems.
so: Ars Comb. 100, 239-256 (2011).
py: 2011
pu: Charles Babbage Research Centre, Winnipeg, MB
la: EN
cc: 
ut: triple system
ci: Zbl 0336.05008
li: 
ab: A Steiner triple system of order $n$ is a pair $(S,T)$ where $\vert S\vert
    =n$ is a set and $T$ is a collection of triples of points of $S$ such
    that any pair of points of $S$ is contained i exactly one triple from
    $T$. It is well known [{\it T. P. Kirkman},Cambridge and Dublin Math.
    J. 2 191‒204 (1847)] that the Steiner triple system of order $n$
    exists if and only of $n \equiv 1 \text { or } 3 \text { mod } 6$. For
    such integer $n$, denote $Int(n)$ the set of all $k$ such that there
    exists a pair of Steiner triple systems of order $n$ having $k$ triples
    in common. One can observe that $Int(n) \subseteq \{0,1,2,\ldots
    ,n(n-1)/6=x\} \setminus \{x-1,x-2,x-3,x-5\} = I(n)$. {\it C. C.
    Lindner} and {\it A. Rosa} [Can. J. Math. 27, 1166‒1175 (1975; Zbl
    0336.05008)] proved that $Int(n)=I(n)$ holds for every $n \equiv 1
    \text { or } 3 \mod 6$, except for $n=9$. A $λ$-fold triple system of
    order $n$ is a pair $(S,T)$ with $\vert S\vert =n$ and $T$ a collection
    of triples of points of $S$ such that any pair of points is covered by
    $λ$ triples. A hexagon triple is formed by three triples such that any
    pair of them intersect in one vertex but all of them have the empty
    intersection (they can be viewed as consecutive triples along a
    hexagon). The paper focuses on hexagonal triple systems, a special type
    of $3$-fold triple systems which can be partitioned into hexagon
    triples. The author gives a complete solution of the intersection
    problem for hexagonal triple systems modulo a few possible exceptions
    for $n=13$.
rv: Ondřej Pangrác (Praha)