Gemara

Mishnah (73b): If one eats the volume of a big date including its pit on Yom Kipur he is liable.

79a - Rav Yehudah: This is more than k'Beitzah.

79b - Rav Zevid: It is less than k'Beitzah.

Source (Mishnah - R. Meir): One must eat a k'Zayis to (bless Birkas ha'Mazon and) be included in a Zimun;

R. Yehudah says, he must eat a k'Beitzah.

They argue about how to expound "V'Ochalta v'Sovata". R. Meir says that v'Ochalta refers to eating (this normally connotes a k'Zayis) and v'Sovata alludes to drinking;

R. Yehudah expounds v'Ochalta v'Sovata to mean eating that gives satiation, i.e. a k'Beitzah.

Surely, if a k'Beitzah satiates, it relieves one who was fasting!

80a - R. Avahu: "Mi'Kol ha'Ochel Asher Ye'achel" - Chachamim determined that one can swallow a k'Beitzah at once.

Kerisus 14a - Mishnah: One can be liable four Chata'os and one Asham for one eating, e.g. if a Tamei ate Chelev that was Nosar of Kodshim on Yom Kipur.

Question: Why didn't the Mishnah list a fifth Chiyuv Chatas, for Pigul?

Answer #1: The Tana discusses eating only one k'Zayis.

Question: To be liable for eating on Yom Kipur, one must eat a large date, which contains two k'Zeisim!

Answer: The case is, he ate other food as well. The k'Zayis of Chelev completed the Shi'ur to be liable for Yom Kipur.

Answer #2 (Rav Ada bar Acha): The text of the Mishnah says that one is liable five Chata'os (and an Asham), i.e. he is also liable for a k'Zayis of Pigul.

Question: If so, he can be liable a sixth Chatas, for eating blood!

Answer: The Tana teaches the Chiyuvim that can come for one swallowing. One cannot swallow more than two k'Zeisim at once.

Shabbos 91a - Question (Rava): If one threw a k'Zayis of Terumah into a Tamei house, what is the law?

He asks about Shabbos. The case is, less than k'Beitzah was in the house, and the k'Zayis landed on it. Since it completes the Shi'ur of k'Beitzah for Tum'ah (it has importance like k'Grogeres), so he is liable for Hotza'ah;

Or, perhaps a proper k'Grogeres is required?

Rishonim

Rambam (Hilchos Shevisas Asor 2:1): If one eats food fitting for people, and it is the volume of a big date, which is slightly less than k'Beitzah, he is liable.

Source (Ran 3a DH Masnisin b'Pirkin): Amora'im argue about whether the date is bigger or smaller than a k'Beitzah. Presumably, the opinions are close to each other, so Rav Zevid holds that it is only slightly smaller.

Question: The Mishnah says 'and its pit'. Why did the Rambam omit this?

Answer (R. Mano'ach): It is understood that the Shi'ur includes the pit, just like a k'Zayis, regarding which the pit is not mentioned. The Yerushalmi concludes that the Mishnah mentioned the pit to exclude the air pocket inside. The Rambam teaches that the Shi'ur is slightly less than k'Beitzah, so he did not need to discuss the pit and the pocket.

Rosh (8:11): If one eats the amount of a large date and its pit he is liable.

Poskim

Shulchan Aruch (OC 612:1): If one eats the amount of a large date he is liable. this is slightly less than k'Beitzah.

Mishnah Berurah (2): This refers to an average size egg. Acharonim argue about whether or not it includes the shell.

Taz (1): The Tur (and Shulchan Aruch) only bring Halachos that are relevant today. The Shi'ur for Chiyuv is relevant for a sick person who must eat on Yom Kipur. If possible he should not eat enough to join for a Shi'ur.

Shulchan Aruch (486:1): Some say that the Shi'ur of a k'Zayis is about half a k'Beitzah.

Question: Kerisus 14a says that a date is two k'Zeisim, and the Shulchan Aruch (612:1) says that it is less than a k'Beitzah. If so, k'Zayis is less than half k'Beitzah! The Shulchan Aruch (368:3) says that a fig is three or four ninths of k'Beitzah, yet the Gemara (Shabbos 91a) says that it is bigger than a k'Zayis. If so, a k'Zayis is less than half a k'Beitzah!

Answer (Magen Avraham): The Shulchan Aruch is stringent here (to say that one should eat a half k'Beitzah for the k'Zayis of Matzah).

Question: In Kerisus (14a( it says that a person can swallow two k'Zeisim at once. Elsewhere, the Gemara says that one can swallow a k'Beitzah at once!

Answer #1 (Magen Avraham): The Gemara in Kerisus means that one cannot be liable for blood in the same swallowing, for one cannot swallow three k'Zeisim at once, but one can swallow more than two k'Zeisim at once.

Question: The Rambam (Hilchos Eruvin 1:9) holds that a fig is a third of k'Beitzah. Since a fig is more than a k'Zayis, this means that one can swallow more than three k'Zeisim at once!

Answer #1 (Magen Avraham): Shabbos 91a discusses a small olive. Elsewhere, the Gemara discusses average olives.

Objection (Noda bi'Yehudah OC 1:38): Shabbos 91a discusses Tum'ah. Surely, the Shi'ur is an average olive!

Answer #2 (to questions (iii) and (v) - Noda bi'Yehudah ibid.): Tosfos Yeshanim (Kerisus 14a DH Ein) explains that a person can swallow two k'Zeisim at once 'Derech Achilah'. I.e., even though a person can force himself to swallow up to an egg (which is three figs, more than three k'Zeisim), this is Achilah Gasah. The Gemara in Kerisus discusses liability for eating, which is only Derech Achilah.