Three-dimensional stable matching with cyclic preferences

We consider stable three-dimensional matchings of three genders (3GSM). Alkan [Alkan, A., 1988. Non-existence of stable threesome matchings. Mathematical Social Sciences 16, 207–209] showed that not all instances of 3GSM allow stable matchings. Boros et al. [Boros, E., Gurvich, V., Jaslar, S., Krasner, D., 2004. Stable matchings in three-sided systems with cyclic preferences. Discrete Mathematics 286, 1–10] showed that if preferences are cyclic, and the number of agents is limited to three of each gender, then a stable matching always exists. Here we extend this result to four agents of each gender. We also show that a number of well-known sufficient conditions for stability do not apply to cyclic 3GSM. Based on computer search, we formulate a conjecture on stability of “strongest link” 3GSM, which would imply stability of cyclic 3GSM.