On Triangular Diophantine Sets

Let $m\geq 2$ be an integer. For integer $n \neq 0$, we introduce one variation of Diophantine problem, when we call the set of $m$ distinct
positive integers a Triangular $D(n)$-$m$-tuple if any product of its two distinct elements increased by $n$ is a triangular number. As usual
in similar problems, the question of interest asks the cardinality of such Triangular Diophantine sets. In this talk we will present some work on
the extendibility of various Triagular triples and give some ideas for future search. It is joint work with L. Szalay, Y. Fujita and A. Jurasić.