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Given $n$ men, $n$ women, and $n$ dogs, each man has an incomplete preferencelist of women, each woman does an incomplete preference list of dogs, and eachdog does an incomplete preference list of men. We understand a family as atriple consisting of one man, one woman, and one dog such that each of thementers in the preference list of the corresponding agent. We do a matching as acollection of nonintersecting families (some agents, possibly, remain single).A matching is said to be nonstable, if one can find a man, a woman, and a dogwhich do not live together currently but each of them would become "happier" ifthey do. Otherwise the matching is said to be stable (a weakly stable matchingin 3-DSMI-CYC problem). We give an example of this problem for $n=3$ where nostable matching exists. Moreover, we prove the absence of such an example for$n<3$. Such an example was known earlier only for $n=6$ (Biro, McDermid, 2010).The constructed examples also allows one to decrease (in two times) the size ofthe recently constructed analogous example for complete preference lists (Lam,Plaxton, 2019).

Minimal instances with no weakly stable matching for three-sided problem with cyclic incomplete preferences