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Sukka 3a2

The Braisa says that a house less than four Amos squared is not a house. Therefore it's not obligated to put up a Mezuza, or to put up a fence on the roof. It's not Tamei if it has Tzaras and if it's sold in a walled city, it still could be redeemed after a year. Those who live there don't need to contribute to the Eiruv, Shituf (an Eiruv for an alleyway) and you cannot put the Eiruv in there.

Tosfos says (this definitely fits into the Shita that an Eiruv works by being part of the household. People live where their food is and therefore they all considered living in that house. If the house is not livable because it's too small, then you don't have an Eiruv.)
But according to Shmuel that holds that Eiruv works as a buying, it would seem that this Halacha doesn't fit. (The food has a worth, so it acts like money. We assume Eruv works that everyone gives some food to buy a stake in the other people's share of the courtyard.) It would seem that you can acquire each other's share in the courtyard no matter if the house you put the Eruv is livable or not.
Tosfos says this is not the true way to understand Shmuel. It's not that you all acquire a stake in the other's share of the courtyard. Rather, by giving the food, you acquire a stake in the house which you place the food. For receiving a stake of the house, they give over their stake of the courtyard to the owner of the house. But they would only give over their stake of the courtyard for a stake in a livable house, but not one less than four Amos.
Tosfos proves that this is the proper way to understand Shmuel. Since Shmuel says that the owner of the house doesn't need to contribute to the Eruv, since everyone is living in his house. We see that even Shmuel holds that the acquiring is done is for a stake in the house, and not just acquiring the other's share in the courtyard.

The Gemara says that you also cannot combine two cities (to be considered one city for T'Chumim) with this house. Nor do brothers that split an estate cannot split it. (We'll explain this later.)

Tosfos brings from the Yerushalmi that produce that's brought into it is not yet obligated to take off Maaser (as if it was a real house.) Also, those that made an oath not to go into a house, is permitted to go into such a house. Tosfos gives another possible Nafka Minah. If a Naarah M'orasah (a married young girl) who is Mizana and gets Skila, it's done by the door of her father's house if he has one. If it less than four Amos, this cannot be considered her father's house, so you don't do Skila there.

Sukka 3a1

The Gemara says the Halacha is like Bais Shamai that the smallest Sukka must hold the head and most of a person with his table. Not like Bais Hillel that doesn't need it to be big enough for the table.

Tosfos quotes R' Amram that says we Paskin like Bais Shamai 6 times. One time is here by Sukka. Three of them are by Tzitzis. Bais Shamai holds that linen clothes are exempt from Tziztis while Bais Hillel obligates them. Bais Shamai says you need four strings and Bais Hillel says you need three. Bais Shamai says the strings must be four long while Bais Hillel holds three.
The other two is in the eight Perek of Brachos. Bais Shamai holds that you sweep the floor before washing Mayim Achronim and Bais Hillel holds you may wash first. Bais Shamai holds if you leave the place you ate, you must return to that place to Bentch while Bais Hillel says you may Bentch wherever you are. We learn that we should do the latter like Bais Shamai from the stories that some went back like Bais Shamai and they found a treasure, while another remained in his place and was eaten by a lion.
Although the Gemara says the Halacha in the whole Perek is like Bais Hillel (infers even in the case of not returning to the place of eating) except for washing before sweeping, since Bais Hillel agrees that it's better to go back, therefore you should.

The Gemara asks how do we know that Bais Shamai and Bais Hillel argue in how big is the smallest Sukka? Maybe they only argue in a big Sukka if you need to have your table inside the Sukka or not. Bais Shamai says you cannot have your table in the house, lest you are drawn towards it and you'll be eating outside the Sukka. This seems to be inferred in the Mishna there that says "if they had the table in the house" and does not say "if the Sukka cannot hold the table."

The Gemara answers that this is the inference from two Braisas. Both of them bring the Shita of Rebbe that a Sukka needs to be 4 Amos wide. The difference between them is how they bring the Rabanan. One quotes them as saying it needs to fit his head and most of his body. The other Braisa adds the table. We must say that the first Braisa is like Bais Hillel and the second is like Bais Shamai.

Furthermore, the Mishna itself infers they're arguing about the width of the Sukka, since it says that Bais Shamai says the Sukka is Pasul and Bais Hillel says it's Kosher. If they're only arguing if you may keep the table in the house, then they should argue if the person is Yoitze or not, since there is nothing wrong about the Sukka. Though we already said the Mishna infers that they're arguing with having the table in the house, we must conclude that they're arguing in both cases.

Tosfos asks that the Gemara in Brachos says that the Halacha is like R' Yochonon Hachurni that holds like Beis Hillel (that the elders of Beis Shamai and Beis Hillel visited him and he had his table inside the house. The elders of Beis Shamai said that he was never Yoitze Sukka.) But here the Gemara says that the Halacha is like Beis Shamai?
The Gemara over there only Paskin like Beis Hillel by a big Sukka that had the table inside the house. Here, we only Paskin like Beis Shamai by a small Sukka.
The Gemara in Brachos said from that Gemara we see who ever does like Beis Shamai wasn't Yoitze anything. Tosfos asks, what proof does it bring from there about doing like Beis Shamai if the story was about someone doing like Beis Hillel?
Another question: If the Halacha is like Beis Hillel, why do we need a proof if you do like Beis Shamai you weren't Yoitze?
Tosfos answers that we're bringing a proof to a case where Beis Hillel makes a Gezeira and Beis Shamai doesn't and leaves the Halacha as it was given form the Torah, if someone does it like Beis Shamai, he's not even Yoitze the Torah Mitzvah. Similar to what Beis Shamai held when R' Yochanan did like Beis Hillel, that they said he wasn't Yoitze the Mitzvah Sukka at all.

Sukka 2b2

R' Huna in the name of Rav says that the Rabanan agree if the Sukka is wider than 4 Amos, it's Kosher even above 20. This is like R' Zeira's Shita. If it's wider than four Amos, then the Schach would be providing the shade.

Tosfos says the Gemara assumes that only to R' Zeira, wider helps to make it Kosher. But according to Rabbah, even though wider should make it more noticeable, it doesn't make it Kosher.
Tosfos asks, if it's wider than four Amos we say that there is shade from the Schach. This infers even if it's a lot higher. But how can being wider than four Amos would make shade no matter how high the Sukka is?
Tosfos answers that it has to be in proportion to four Amos wide to twenty Amos high. If it's taller, then it needs to be even wider.
Another answer, that once it wider than four Amos, no matter how tall it is, it's impossible that it doesn't give some degree of shade. This Tosfos compares to eating on Yom Kipper which the Shiur is universally a Koseves (dry date.) Eating a Koseves calms someone's hunger. To a regular person a lot and to Og Melech Habashon a little bit.

R' Chonon b. Abba says as long as it's wider than 7 T'fachim (the amount that holds the head and most of a person with his table) even the Rabanan agree that's Kosher above 20. This does not follow any of the Shitos that we brought.

Tosfos say he must hold that a 7 T'fuchim Sukka at the height of 20 Amos more resembles a chicken coop than a Sukka. Therefore a little wider makes it look like a Sukka.

The Gemara says the R' Yashiya (that differentiates between the walls reaching the Schach or not) since it doesn't depend on the width of the Sukka, is definitely arguing with the other Shitos. But R' Huna and R' Chonon, maybe all agree that the only Machlokes is at the minimum Shiur Sukka and anything wider is Kosher. They just argue how big the smallest Sukka is. R' Huna says it's four Amos and R' Chonon holds it's 7 T'fachim.

The Gemara counters that everyone could holds that the minimum width is 7 T'fachim. R' Huna says the Machlokes is from 7 T'fachim until four Amos and R' Chonon says the Machlokes is only by 7 T'fachim

Tosfos bring the Girsa of R' Chananal that says that R' Yashiya doesn't argue (i.e. doesn't argue about the width of the Sukka) since he doesn't mention a width.) But let say that R' Huna and R' Chonon who mention different widths argues on how wide the Sukka needs to be etc.

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