A famous conjecture of Ryser states that every $r$-partite hypergraph has vertex cover number at most $r - 1$ times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as $r$-Ryser hypergraphs, have been studied extensively. It was recently proved by Haxell, Narins and Szab�� that all $3$-Ryser hypergraphs with matching number $��> 1$ are essentially obtained by taking $��$ disjoint copies of intersecting $3$-Ryser hypergraphs. Abu-Khazneh showed that such a characterisation is false for $r = 4$ by giving a computer generated example of a $4$-Ryser hypergraph with $��= 2$ whose vertex set cannot be partitioned into two sets such that we have an intersecting $4$-Ryser hypergraph on each of these parts. Here we construct new infinite families of $r$-Ryser hypergraphs, for any given matching number $��> 1$, that do not contain two vertex disjoint intersecting $r$-Ryser subhypergraphs.

8 pages, some corrections in the proof of Lemma 3.6, added more explanation in the appendix, and other minor changes