Eruvin, Chapter 3, Mishnah 11. In the previous Mishnah, Rabban Gamliel listed three stringencies; here we learn three of his leniencies, which are also connected to Yom Tov: "Af hu amar sheloshah devarim l'hakel" - three rulings in which Rabban Gamliel was lenient relative to the Sages. And here too, the halachah does not follow his opinion.
The first leniency - "mechabdin bein hamitos":
In Rabban Gamliel's view, one may sweep the floor between the couches on which people ate on Yom Tov. The basis of the prohibition: one may not sweep a dirt floor, lest a person level out the depressions in the ground - 'mashveh gumos' - and this is the melachah of building, which is forbidden by Torah law. Building means improving the structure of something attached to the ground, and that includes the floor of a house; there is no minimum measure for this melachah, and therefore the prohibition is on a Torah level. Of course, when a person does not intend to do this it is not a Torah prohibition, but it is forbidden in any case.
Rabban Gamliel's argument: we are dealing with a small area that is swept frequently, so a person knows whether a depression has formed in it or not, and it is not common that a depression will be found there right now - therefore it is permitted to sweep it, at least on Yom Tov. On Shabbos even he agrees that it is forbidden, since the punishment for a melachah on Shabbos is more severe than on Yom Tov: on Yom Tov it is lashes, while on Shabbos it is kareis. The halachah was ruled like the Sages, that one may not sweep even on Yom Tov.
In practice nowadays: the widespread custom is that a house whose floor is tiled over the ground may be swept, since there is no concern that depressions will be formed in it.
The second leniency - "u'manichin es hamugmar b'Yom Tov":
The mugmar is the burning of spices: on top of a burning coal one places something that gives off fragrant smoke, such as frankincense, which is dried tree resin. Rabban Gamliel permits this on Yom Tov, because on Yom Tov melachos are permitted for the sake of ochel nefesh - direct bodily enjoyment. And just as one may occupy oneself with eating and drinking, so too one may produce a pleasant fragrance for direct personal enjoyment. Even though there is kindling here, the creation of smoke and the burning of the spice, Rabban Gamliel holds that it is permitted under the principle of ochel nefesh.
In truth there is an additional concern here: not only is there kindling involved, but also extinguishing - for when he places the mugmar on the burning coal beneath it, he thereby extinguishes it.
The Sages said: the leniency of ochel nefesh on Yom Tov applies to things that are 'shaveh l'chol nefesh', that an ordinary person enjoys. Burning incense is not such a thing - only the wealthy and the pampered, like Rabban Gamliel, enjoy it, and most people do not burn spices in their homes after eating. Since it is not equally enjoyed by all, it may not be permitted on Yom Tov, and so is the halachah. This question echoes in the dispute among the later generations regarding smoking cigarettes on Yom Tov.
The third leniency - "v'osin g'di mekulas b'leilei Pesachim":
Rabban Gamliel lived immediately after the destruction of the Temple. When the Beis HaMikdash stood, there was a special manner of roasting the Pesach offering on the night of the Seder: it was roasted on a spit, and Scripture requires that the entire animal be roasted, including the legs and the innards. The legs had to be cut off so that the roasting would be done properly and they would not burn on the spit, and the intestines had to be removed in order to clean out the waste inside them, since they are not cooked as they are.
Therefore the legs and the intestines were placed on the spit, with the head of the roasting animal above them. Chazal saw in this a resemblance to a soldier carrying his weapons and his helmet, since everything is placed together - and from here comes the term 'mekulas', like one who is ready for war or who has a helmet on his head. This was the tradition they had for roasting their meat on the night of the Seder.
Even after the Destruction, when there was no longer a Pesach offering, everyone still longed for meat roasted in that special way, and Rabban Gamliel was lenient and said that it may be done even nowadays: this is not an offering at all but ordinary meat, only that it is roasted in the old way that everyone loved.
The Sages, however, forbade it: people will see this meat roasted in the form of the Pesach offering and will think that offerings of Pesach are indeed being consecrated and eaten outside Jerusalem - something that cannot be done while the Beis HaMikdash lies in ruins. Because of this concern that people will err, they forbade roasting a gedi mekulas - a lamb or a goat prepared in the way the Pesach offering was roasted - on the night of the Seder.
The Shulchan Aruch writes that if the meat is cut into pieces the way ordinary meat is prepared - something like shishlik roasted on a skewer - it is permitted. The Gemara, though, limits this leniency to a place where the custom is to eat roasted meat on Pesach. And the Mishnah Berurah writes that throughout the lands of Ashkenaz the custom is not to eat roasted meat at all on the night of the Seder - and this is our custom, the custom of Ashkenazim - so that it should not enter people's minds that there is any resemblance to the offering and they come to err.
In summary: In this Mishnah, Rabban Gamliel listed three leniencies: sweeping between the couches, which is forbidden because of leveling hollows, a form of the labor of building; bringing in the incense burner, which was forbidden because it is not equally enjoyed by everyone; and preparing a gedi mekulas on the night of the Seder, which was forbidden out of concern that people would err and think that the Pesach offering is brought nowadays. In all three, the halachah was decided according to the Sages and not according to Rabban Gamliel.