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)