Chapter 3, Mishnah 13. Our mishnah lists three areas in the laws of Yom Tov in which Rabban Gamliel ruled stringently, while the halachah was decided in accordance with the lenient view of the Sages. The source of this passage is the mishnah in Maseches Beitzah, chapter 2, mishnah 6, which reads: "Rabban Gamliel machmir k'divrei Beis Shammai" - Rabban Gamliel was stringent in accordance with the view of Beis Shammai.
It is worth pausing over how remarkable this is: Rabban Gamliel was the grandson of Rabban Gamliel the Elder, who was the grandson of Hillel the Elder. Even though he came from the dynasty of Hillel and served as Nasi, in these three areas he ruled stringently like Beis Shammai.
The first stringency - insulating food on Yom Tov for Shabbos:
The underlying principle: on Shabbos one may not cook at all, and on Yom Tov one may cook only for the needs of Yom Tov itself and not for the following day - even when the following day is Shabbos. Since a person who does not cook on the eve of Shabbos that falls on Yom Tov would be left with no food for Shabbos, the Sages ruled that one may cook on Yom Tov for Shabbos, provided that he began the preparation before Yom Tov entered. This is the eruv tavshilin, whose whole point is that the preparation for Shabbos began in advance and was not left to the last moment.
As is our practice today, the eruv tavshilin consists of two items:
A baked item - made with dry heat, usually bread.
A cooked item - made with water, usually a hard boiled egg.
These are only examples, but this is how it is actually done. According to Beis Hillel, this eruv covers all the cooking needs of the eve of Shabbos that falls on Yom Tov.
Beis Shammai hold that insulating food is an entirely separate matter. The Sages forbade insulating food on Shabbos itself, either because it looks like cooking or out of concern that one may come to stir the coals. Therefore, in their view one may not insulate food on Yom Tov for Shabbos unless a separate act of insulation was performed on the day before Yom Tov, as an additional component of the eruv tavshilin. The halachah, however, follows Beis Hillel: a baked item and a cooked item are enough to permit insulating food as well.
The second stringency - assembling a lamp on Yom Tov:
One may not assemble on Yom Tov the separate parts of a lamp that comes apart, such as a lantern. On Shabbos one does not deal with lamps at all, since nothing may be lit in them, whereas on Yom Tov one may light a flame from an existing fire. As for assembling the lamp itself, Beis Hillel hold that when the object is a utensil that is normally taken apart and put back together - like a travel candle set made of units that fit into one another - taking it apart is not considered breaking and putting it together is not considered repairing, since this is simply how it is used, and therefore it is permitted.
Beis Shammai hold that there is a concern here of the melachah of building, since in their view the concept of building applies to utensils. One who joins the separate units to form the lamp, that is, the candle holder, is considered to be building, and therefore it is forbidden. Rabban Gamliel explicitly agrees with this. But the halachah follows Beis Hillel, that there is no building with regard to utensils, and therefore it is permitted.
The third stringency - "v'ein ofin pittin g'ritzin, ela r'kikin" - one may not bake thick loaves, only thin wafers:
Understood simply: one may not bake thick loaves of bread on Yom Tov, but only thin wafers, something like a wafer or a cracker-like bread. But the problem does not lie in the shape of the bread; it lies in its quantity, and the issue is excessive exertion on Yom Tov. Even though cooking for the needs of Yom Tov is permitted, excessive effort is forbidden. And not only is kneading a large quantity of thick dough more laborious than kneading thin dough, but baking "pitin gritzin" - those large loaves - carries a further problem within it: in the end, leftover bread will remain after Yom Tov, and it turns out that he has performed labor on Yom Tov for the sake of a weekday.
Therefore Beis Shammai forbid it, and Rabban Gamliel agrees with them. Beis Hillel, however, and this is the halachah, hold otherwise: true, not all the bread will be eaten on Yom Tov, but the taste of the bread is far superior when the oven is full of loaves baking together. Since that is so, even though part of the bread will be used after Yom Tov, there is a benefit on Yom Tov itself in terms of the quality of the taste, and this is not excessive exertion - it is not effort without good reason, but effort for an excellent purpose. That is enough to permit the matter on Yom Tov for one who wishes to do so, and this is the position of Beis Hillel, unlike Rabban Gamliel.
On this last point Rabban Gamliel said: "Mi yodei'a shel beis abba" - that is, the entire tradition of his father's house, the house of the Nasi from generation to generation, testifies to this. His father, Rabban Shimon ben Gamliel, who was among the last of the Nesi'im before the destruction and was counted among the Ten Martyrs, never had large loaves of bread baked in his home on Yom Tov, but only thin wafers and in a small quantity.
The Sages answered him: "U'mah na'aseh l'veis avicha, shehayu machmirin al atzman u'mekilin al Yisrael" - even if this is accurate, it is no proof and it teaches us nothing. True, in the house of the Nasi those loaves of risen bread were not baked on Yom Tov, but that was a stringency they took upon themselves out of their piety; for the rest of Israel, however, they permitted baking "pitin gritzin" with all the additional labor involved, and even "chori". They were leaders, and they understood that the matter is permitted by the letter of the law, and they did not seek to restrict the public.
And what is "chori"? It comes from the word chor, a hole, like a hole in the ground: bread baked over coals in a hole in the ground. Apparently there were people who loved this food and requested it especially in honor of Yom Tov, but it involves a great deal of exertion - one must keep stirring up the coals and keep the fire burning so that the bread will bake over them. Even so, the tradition of the house of the Nasi itself, the house of Rabban Shimon ben Gamliel, was that he permitted all of Israel to bake even "chori", that labor-intensive bread.
The halachah in this case does not follow Rabban Gamliel, but the Sages.