Machshirin, Chapter 2, Mishnah 5. This mishnah places us in a city whose residents are mixed, part Yisrael and part nochrim, and it resolves a practical question by looking at which group forms the majority of the town.
The case: "ir sheYisrael venochrim dorim bah", a city inhabited by both Jews and non-Jews, "vehayesah bah merchatz haromeches beShabbos", and in that city there was a bathhouse whose waters were kept hot and which stayed open on Shabbos. The question is whether a Jew may go and bathe there once Shabbos has ended.
"Im rov nochrim, rocheitz miyad": if most of the city's population is non-Jewish, one may bathe there right away. "Ve'im rov Yisrael, yamtin kedei sheyeichamu hachamin": if most of the city is Jewish, one must wait the length of time it would take to heat the water.
The decree of kedei sheya'asu
Behind this ruling stands a rabbinic decree commonly called kedei sheya'asu. A non-Jew is of course not commanded to rest on Shabbos, and his own work is not a transgression. But when he performs a melachah on Shabbos on behalf of a Jew, the Chachamim ruled that the Jew may not derive benefit from that work. This applies on Shabbos itself, and it does not end when Shabbos ends: once Shabbos is over, the Jew must let enough time pass that the melachah could have been performed after Shabbos before he takes any benefit from it.
Our mishnah applies precisely this rule. Where Jews form the majority of the town, the presumption is that the bathhouse water was warmed for their sake, and so a Jewish resident delays after Shabbos until as long a stretch has passed as heating the bath fresh would require, and only then does he enter to bathe.
Half and half
"Mechtzah al mechtzah, yamtin kedei sheyeichamu hachamin." If the two populations are equal, the same waiting period applies. A Jewish population of half the city is a significant presence; it is not swallowed up and nullified in a larger group of non-Jews, since here there simply is no non-Jewish majority. We therefore treat the heating as having been done for Jews as well, and one must wait after Shabbos for the time it would take to heat the water.
The opinion of Rabbi Yehudah
"Rabbi Yehudah omer: be'ambati ketanah, im yesh bah rashus, rocheitz bah miyad." Rabbi Yehudah speaks of a small pool. If there is a rashus, a person of importance, in the city, one may bathe in that small pool immediately.
The Rav points out that a rashus is not necessarily a king. The term covers any distinguished figure who has aides and household staff attending to his wants. Once such a personage lives in the town, we presume the little pool is kept for him, its water held warm so that it stands ready the moment he wishes to use it. Under those circumstances, a Jew may bathe the instant Shabbos departs, without any delay whatsoever, and this holds true even in a town whose residents are mostly Jewish, since the presumption is that the water was warmed not for the Jewish residents at all but for this eminent official.