Loops with Partitions and Matchings

An alternating circular list of distinct \(r\)-element subsets of some finite set \(X\) and distinct \(r\)-partitions of type \(\tau\) is said to be a \(\tau\)-loop if successive members of the list are orthogonal. We address the problem of finding complete \(\tau\)-loops including all \(r\)-element subsets of \(X\), for any fixed \(|X|\) and type \(\tau\).