A collection of $6$-tuples on 18 points with the property that each pair is covered $5$ times is a balanced incomplete block design with $(v,k,\lambda) = (18,6,5)$ and $t=2$. The condition that you can schedule the matches to occur simultaneously in $17$ rounds is that the design is resolvable.
This article claims to construct a resolvable block designs with parameters including $(18,6,5)$.
A collection of $6$-tuples on 18 points with the property that each pair is covered $5$ times is a balanced incomplete block design with $(v,k,\lambda) = (18,6,5)$ and $t=2$. The condition that you can schedule the matches to occur simultaneously in $17$ rounds is that the design is resolvable.
This article claims to construct a resolvable block designs with parameters including $(18,6,5)$.