Given a 3-uniform tripartite hypergraph G(n,m,k), we define a simple cycle to be any set of edges such that each set of two points within every hyperedge is contained in exactly two hyperedges, and the set is connected (where to move from one vertex to another they must be part of hyperedges which share two points.
What is the size of the largest simple cycle in G(n,m,k)?
Every edge must be hit exactly twice, so we have an obvious upper bound of $2*\min(n(m+k),m(n+k),k(n+m))$. Is this bound tight?
Perhaps an easier and more approachable question would be, consider G(n,n,n), is there a simple cycle which is $\Omega(n^2)$?