Gemara

Rav: If Reuven and Shimon jointly bought (before Yom Tov, to divide on Yom Tov) a barrel (of wine) and an animal, and they have different Techumim, the barrel is permitted (each may take his share whenever he may go) and the animal is forbidden (outside of the overlap of their Techumim);

Shmuel: Both (the barrel and animal) are forbidden.

Question: If Rav holds that Yesh Breirah, he should permit both of them (it is as if Reuven's share was his from the start on Yom Tov, so it received his Techum). If he holds that Ein Breirah, he should forbid both of them! (Reuven took a share in exchange for his true share. Perhaps what he took was Shimon's at the start on Yom Tov, and it received Shimon's Techum.)

Answer: Really, he holds that Yesh Breirah. An animal is different, for Reuven's share nourished from (Shimon's, which has) a different Techum, and vice-versa.

Question (Rav Kahana and Rav Asi): (Reuven had no intent to eat Shimon's share; it is Muktzah for him.) If we are not concerned for (the nourishment Reuven's share received from) Muktzah, all the more so we are not concerned for (nourishment from other) Techumim!

Rav did not answer.

Question: What was the conclusion?

Answer: R. Oshaya holds that Yesh Breirah, and R. Yirmeyah holds that Ein Breirah.

Question: Elsewhere, R. Oshaya holds that Ein Breirah!

38a - Answer: R. Oshaya holds that Ein Breirah regarding mid'Oraisa laws. Regarding mid'Rabanan laws he holds that Yesh Breirah.

The Halachah follows R. Oshaya.

Shmuel: A Shor Shel Petem (an ox that is being fattened for sale) is like everyone's feet (on Yom Tov. Whoever buys it, even one from another city who came here through an Eruv, may take it wherever he is permitted);

A shepherd's animals are like the feet of people of his city. (He sometimes sells them to neighbors or friends. A buyer from his city can take it where he himself may go. The shepherd had no intent for people of other cities.)

Rishonim

Rif: Rav holds that Yesh Breirah, but regarding an animal, because the limbs nourish from each other, it is as if the Techumim are mixed, so we cannot use Breirah. Shmuel holds that Ein Breirah in all cases. We conclude like R. Oshaya, that for mid'Rabanan laws Yesh Breirah. This is like Rav, so the Halachah follows Rav.

Rebuttal (Ba'al ha'Ma'or): Rashi explains that only Rav is concerned for nourishment of one part of the animal from the rest. R. Oshaya holds that Yesh Breirah for mid'Rabanan laws, both for a barrel and an animal. Regarding a Shor Shel Petem the Halachah follows Shmuel, who permits. Rav was challenged by Rav Kahana and Rav Asi, and he was unable to answer. This shows that the Halachah does not follow Rav.

Defense (of Rif - Milchamos Hash-m): R. Oshaya is concerned for nourishment of one part of the animal from the rest, just like Rav. The Rif holds that Rav did not see a need to answer them. His silence does not show that he admitted.

Rebuttal (Ran DH Tanu Rabanan): Rav Kahana and Rav Asi were Talmidei Chaverim of Rav. He would not have ignored their question if he had an answer!

Question: Here Shmuel forbids a jointly owned animal, but he permits a Shor Shel Petem and a shepherd's animals!

Answer #1 (brought in Ba'al ha'Ma'or 21a DH v'Zeh): Shmuel said that a Shor Shel Petem is like everyone's feet, i.e. it is forbidden, just like a garment borrowed by partners.

Answer #2 (Ba'al ha'Ma'or): It is better to explain that Shor Shel Petem (or a shepherd's animal) is permitted because the owner is ready to sell them. A shepherd's animal is permitted like most of the people of the city, who did not Me'arev. In such a case there is no one to block the sale, even Shmuel holds that Yesh Breirah for mid'Rabanan laws.

Answer #3 (Milchamos Hash-m): A Shor Shel Petem does not acquire Shevisah at all. The owner plans to sell it, so it does not get his Techum; no one else acquired it at the start of Yom Tov. It is like a Hefker animal; nourishment from different Techumim does not apply. A shepherd's animal did not acquire the Shevisah of any individual. Even if the entire city was Me'arev in one direction, the animal has 2000 Amos in every direction. Alternatively, it has the Techum common to everyone of the city, including those who did and did not Me'arev.

Answer #4 (Hasagos ha'Ra'avad): Perhaps the Shor Shel Petem was slaughtered before Yom Tov, so this does not contradict what Shmuel says about the partners' animal. Perhaps Shmuel holds that there is Breirah regarding foods, but not regarding liquids because they constantly mix. We hold that for mid'Rabanan laws there is Breirah regarding foods and liquids.

Ran (ibid.): Since we hold that Yesh Breirah regarding mid'Rabanan laws, if someone from the city bought from the shepherd it is like his feet.

Rambam (Hilchos Yom Tov 5:20): If Reuven and Shimon jointly bought an animal, and slaughtered and divided it on Yom Tov, all the meat is permitted only in the overlap of their Techumim. However, if they bought a barrel and divided it on Yom Tov, each may take his share wherever he may go. Since Techumim is mid'Rabanan, Yesh Breirah. We cannot say this regarding an animal, for each share nourished from the other; every limb contains parts of both shares.

Rosh (5:7): It appears that the Gemara asked (what was the conclusion?) regarding an animal, because Rav did not answer the question of Rav Kahana and Rav Asi. If the question was about Breirah, we know the general rule that the Halachah follows Rav against Shmuel in Isurim! The Gemara did not say that the Halachah follows Rav. This shows that R. Oshaya disagrees with Rav; even an animal is permitted.

Rosh (8): A Shor Shel Petem is like the feet of whoever buys it. It is like water of a well for Olei Regalim. A shepherd's animals are like the feet of people of his city.

Poskim

Shulchan Aruch (OC 397:10): If Reuven and Shimon jointly bought an animal, and slaughtered and divided it on Yom Tov, all the meat is permitted only in the overlap of their Techumim.

Taz (2): This is because Bein ha'Shemashos each share nourished from (the other share, which has) a different Techum.

Beis Yosef (DH Shnayim): We follow the Rif and Rambam against the Rosh.

Bi'ur Halachah (DH Harei): Most Rishonim hold like the Rosh. Therefore, one may be lenient if there is a great need.

Shulchan Aruch (ibid.): However, if they bought a barrel and divided it on Yom Tov, each may take his share wherever he may go.