Extension to the Stable Marriage Problem ?
There is an open conjecture regarding 3 types of people. Suppose you have men women and dogs so that men have preference lists on women, women have preference lists on dogs, and dogs have preference lists on man. Is there always a stable marriage? 

(For other preference structures on 3-types society the answers are known to be negative).

Another comment is that stable marriage represent a non empty core and there is a well known condition by Scarf that implies the existence of nonempty core. It is known that Scarf conditions are satisfied for the original stable marriage problemand for the house allocation problem. (But failed for the men/women/dogs problem).

Some references: 


A reference to Scarf's paper: H.E. Scarf, The core of an $N$ person game, Econometrica 35 (1967) 50--69.
A paper showing various applications for Scarf's crucial lemma and quote quite a few others: (In particular, a fractional version of Gale-Shapley theorem for hypergraphs by Aharoni and Holzman is described): R. Aharoni, and T. Fleiner, On a lemma of Scarf, J. Combin. Theory Ser. B 87 (2003), 72--80. 
A solution of the men-women-dogs problem when there are at most 4 of each gender appears in a paper by Eriksson et al (Math Soc Sci 2006).