Backdoor (Eruvin 67b)

(a) (Rav Yehudah): If a Nochri has an opening four by four [Tefachim] to a valley, even if he brings camels and wagons [into the Mavoy], he does not forbid the Benei Mavoy.
(b) Question: What is the reason?
(c) Answer: He prefers using his own opening [for the valley is less crowded - therefore, it is as if he has no share in the Mavoy].
(d) Question: If a Nochri has an opening to a Karfef, what is the law?
(e) Answer (Rav Nachman bar Ami): I have a tradition, even if it is open to a Karfef [he does not forbid].
(f) (Rabah and Rav Yosef): If a Nochri has an opening to a Karfef [at most] Beis Sa'atayim, he forbids (it is not so big, he still uses the Mavoy); if it is more than Beis Sa'atayim, he does not forbid;
(g) If a Yisrael has an opening [to a Karfef at most] Beis Sa'atayim, he does not forbid (it is big enough for his needs on Shabbos); if it is more than Beis Sa'atayim (it is a Karmelis, he cannot bring from his house to it) he forbids [unless he participates in the Eruv].

Summary courtsey of Kollel iyun hadaf. The parentheses and brackets are their additions and not the translation of the gemara. In (g), they added in the words "at most" but that is not clear that those words should be there. The Sfas Emes assumes there there must be a minimum size to the karfef in order for the Jew not to forbid the others in the chatzeir. He is unsure what that minimum is. He suggests that maybe it is a beis sa'atayim which would mean that the only way he doesn't forbid is if the karfef is exactly that size. He finds that hard to believe but doesn't have any other suggestions for that minimum size.

I'm not sure why he doesn't quote Rashi d"h yisrael who says that the reason it doesn't forbid is because it's bigger than the courtyard. So that seems to be the minimum - as long as it's bigger than the chatzeir. Why is that the minimum shiur? It could be that as long as the Jew has this backdoor we will force him to use it (I think that we don't actually force him to use it but just force him to be mevatel his rights in our chatzeir since he could use that one - he could still walk through our chatzeir as a guest) so that he won't forbid the rest of us as long as it's bigger than the chatzeir because he really doesn't lose much. So it's a classic zeh nehene v'zeh lo chaseir. He might be losing a little because we're forcing him to use the backdoor but as long as it's bigger even by a little it balances out.