Gemara

(Mishnah): If one vowed not to eat Mazon, he is permitted [only] water and salt.

30a (Rav Huna): The Mishnah discusses one who forbade 'anything that is Zan' (satiates).

Inference: Water and salt do not satiate, but everything else does.

Berachos 48a (Rav Yehudah brei d'Rav Shmuel): If nine people ate bread and one ate vegetables, they join [towards 10 to say Hash-m's name in the Zimun].

Questions (R. Zeira): If seven [or eight] ate bread and three [or two] ate vegetables, do they join?

Answer (Rav Yehudah): They join.

R. Zeira: I need not (some texts - I regret that I did not) ask whether four can join to six who ate bread!

R. Yirmeyah: It is good that you did not ask. Two and three join, for the majority ate bread. The same applies when four ate vegetables!

R. Zeira held that a noticeable majority is required.

R. Shimon ben Shetach drank a cup of wine, and blessed Birkas ha'Mazon [for Yanai ha'Melech and his friends].

(R. Aba): This is Shimon ben Shetach's opinion;

(R. Chiya bar Aba): One may not be Motzi others (in Birkas ha'Mazon) unless he ate a k'Zayis of grain.

(Beraisa - R. Shimon ben Gamliel): If one sat with people, even if he ate only vegetables dipped in brine or one fig, he joins [for a Zimun].

48b (Rav Chana bar Yehudah): If one ate a vegetable leaf or drank a cup of wine, he joins for a Zimun.

Rishonim

Rambam (Hilchos Berachos 5:8): We are Mezamen only on one who ate at least a k'Zayis of bread. If seven ate bread and three ate vegetables, brine or similar things, they join to be Mezamen with Hash-m's name. The Mevarech must be one who ate bread. If six ate bread and four ate vegetables, they do not join. Those who ate bread must be an evident majority.

Rosh (Berachos 7:21): Three who ate vegetables join with seven who ate grain, but four do not join with six, for we require an evident majority. Bahag says that one who ate a k'Zayis can be Motzi only those who ate a Shi'ur [to be liable in Birkas ha'Mazon] mid'Rabanan. This is wrong. Had R. Shimon ben Shetach eaten a k'Zayis of grain, all would agree that he could be Motzi Yanai and his colleagues, who ate to satiation. In Rosh Hashanah (29b) we say that one who was Yotzei can be Motzi, just for Berachos before food, one may bless for another only if he himself will eat. One must eat a k'Zayis in order to say she'Achalnu mi'Shelo, but be can Motzi others in Birkas ha'Mazon even without this. However, the Yerushalmi says unlike this, based on a verse. Perhaps this is only mid'Rabanan, and it is only an Asmachta. The ideal Mitzvah is for one who ate to satiation to be the Mevarech. Perhaps the Gemara mentioned one who ate grain to teach that he can be Motzi even if he ate grain that was not made into bread.

Tosfos (48a DH Tish'ah): If nine ate grain and one ate vegetables, they join. This is the Halachah. One who ate vegetables or drank a cup of wine joins. He can even say "she'Achalnu mi'Shelo", for drinking is including in eating. The Yerushalmi says that even two or three who drank join for a Zimun with 10, for this reason. We require only an evident majority.

Poskim

Shulchan Aruch (OC 197:2): If nine ate grain and one ate a k'Zayis of vegetables, they join to mention Hash-m's name.

Beis Yosef (DH v'Chasav): The Mordechai (Berachos 173) says that one who drinks to join must drink a Revi'is. One who eats to join must eat a k'Zayis. The Kolbo says that some say that this refers to joining to a Zimun with three, according to those who require that all three eat bread. For Zimun with 10, for which it suffices to eat vegetables or drink, those who join with the seven need not eat a k'Zayis. Or, perhaps they need not eat bread, but they must eat a k'Zayis of vegetables or drink a Revi'is.

Gra (DH k'Zayis): The Mordechai says that he must eat a k'Zayis. It seems that he learns from "nine who ate..." This connotes the only difference is what they ate, but the Shi'ur is the same. This is difficult, for we say there 'one fig', which is less [some change the text to say "more"] than a k'Zayis, like it says in Shabbos (91a)! We also say 'one leaf of a vegetable.'

Damesek Eliezer: We must rearrange the Gra's words. He supports the Mordechai from the Beraisa that mentions a fig, i.e. it did not discuss a smaller fruit, but he asks from the leaf, and leaves this difficult.

Mishnah Berurah (8): He must eat a k'Zayis in order to say she'Achalnu mi'Shelo. "Achilah" connotes at least a k'Zayis. Alternatively, it is to obligate him to say a Berachah Acharonah.

Magen Avraham (4): If he already said a Berachah Acharonah, he cannot be Mezamen with them, for he already left. This is unlike 194:1. (If three ate bread and one said Birkas ha'Mazon without a Zimun, the others can make a Zimun on him retroactively.) There, he ate bread.

Bi'ur Halachah (DH k'Zayis): The Gra connotes that he does not require a k'Zayis to join to 10. It seems that the Ro'oh agrees. He supported the Rif [who says that he joins only towards 10] from the Mishnah. It discusses a Zimun with three, and requires a k'Zayis. This shows that less than a k'Zayis joins towards 10. Shibolei ha'Leket brought a Tosefta that says less than k'Zayis of grain joins towards 10. Presumably, the same applies to less than k'Zayis of vegetables. However, perhaps even when there is no Berachah Acharonah, grain is more important than vegetables. It is hard to be lenient against the Shulchan Aruch, for presumably all the Poskim who do not distinguish three from 10 agree with the Mordechai and Shulchan Aruch. If one of the people left, the others may count him towards a Zimun of three, but not towards 10! (This shows that it is harder to join towards 10.) Even though the Rashba is more stringent than about three and requires that all ate grain, he requires a Chiyuv Berachah Acharonah to join towards 10. The Tosefta that Shibolei ha'Leket brought is not a proof. It discusses one who dipped in brine. Perhaps together, the brine and vegetables were a k'Zayis!

Shulchan Aruch (ibid.): Even if he merely dipped with them in brine, or drank one cup with a Revi'is of any drink other than water, he joins with them.

Beis Yosef (DH v'Nir'eh): We say that one who drank a cup joins. It seems that this is if he drank anything other than water, for it does not satiate, like it says in Eruvin.

Magen Avraham (5): [One who dipped in brine] must have a Shi'ur.

Eshel Avraham: The brine joins with the vegetables to a k'Zayis.

Magen Avraham (6): Even though water does not satiate, he can say she'Achalnu mi'Shelo, for drinking is included in eating! YD 238:2 connotes like this. (If one swore not to eat, and he drank [even water], he is liable.) Tosfos says that it is because drinking is included in eating. The Gemara mentioned a cup of wine, but it is not limited to wine. The same applies to other drinks. It mentioned wine to teach that even if he drank wine, he cannot [be the Mevarech and] be Motzi others [who ate bread]. All the Poskim wrote Stam that he drank. Therefore, it seems that he joins.

Mishnah Berurah (12): Water does not satiate, therefore, it is not proper to join, even if he was thirsty and wanted to drink [so he must bless before and after]. Some Acharonim bring the Magen Avraham, who says that he joins. Bigdei Yesha, Magen Giborim and Chemed Moshe rule like the Mechaber.

Bi'ur Halachah (DH she'Yesh): The Gra (DH she'Yesh) says that the Mordechai requires a Revi'is based on his opinion, i.e. but the Gra disagrees. I wrote that it is hard to be lenient [about less than k'Zayis]. Eliyahu Rabah says 'Tzedah l'Derech says that the majority of a Revi'is suffices, but I did not see other Poskim agree.' The Gra agrees. Also, the majority of a Revi'is is an important Shi'ur for Kidush. Many Poskim are unsure if one says a Berachah Acharonah even for a k'Zayis of liquid! We are stringent not to bless, but perhaps one may be lenient to join to 10. L'Chatchilah, surely it is better to drink a full Revi'is.

Kaf ha'Chayim (16): He must eat the k'Zayis or drink the Revi'is at the same table with them.

Shulchan Aruch (ibid.): The Mevarech must be one who ate bread.

Gra (DH v'Hu): The Rambam requires that he ate bread. The Rosh was unsure about this.

Mishnah Berurah (13): Since the Mevarech is Motzi the others in their obligation of a Zimun, this is more stringent.

Shulchan Aruch (ibid.): Even if seven ate grain and three ate vegetables, they join, but not if six ate bread and four ate vegetables, for we require an evident majority.

Beis Yosef (DH u'Mah): Rashi explains that R. Zeira regretted that he did not ask about six, for perhaps we require an evident majority. R. Yirmeyah said that there was no need to ask, for a majority suffices. All the Poskim require an evident majority. According to Rashi, why do we abandon what was clear to R. Yirmeyah, and follow the Safek of R. Zeira?! I say that the Poskim explain 'he held that an evident majority is required' to say that R. Zeira was sure. Therefore, he did not ask. The Agur brings from Or Zaru'a that four who ate vegetables join. He explains like Rashi, and rules like what was clear to R. Yirmeyah. Surely we ignore Or Zaru'a's lone opinion against all the Poskim.

Kaf ha'Chayim (17): The Aruch ha'Shulchan says that the Ba'al ha'Ma'or holds like the Agur, and so say the Yerushalmi and Medrash Rabah. However, the Acharonim require seven who ate bread.