Gemara

(Mishnah): If Reuven was Mudar Hana'ah from Shimon and they were alone on the road, and Reuven has nothing to eat, Shimon leaves the food on a rock and declares it Hefker (ownerless). Reuven may take it;

R. Yosi forbids this.

(R. Yochanan): R. Yosi holds that just like one may retract from a gift until the recipient gets it, one may retract from Hefker until someone takes it.

44a (Beraisa #1): If Reuven declared his field to be Hefker, for three days he may retract, but not after this. If he said 'this field is Hefker for one day', or week, month, year or Shemitah cycle, he can retract only until someone acquires it.

Question (Beraisa #2): If one made his vineyard Hefker and promptly harvested it, he must leave Leket, Shichchah and Pe'ah (gifts for the poor). He is exempt from Ma'aser.

If it is not yet Hefker, why is it exempt from Ma'aser?

45a - Answer: In Beraisa #2, he made it Hefker in front of three people. R. Yosi considers this to be proper Hefker. In Beraisa #1, Hefker was done in front of two. This is not Hefker mid'Oraisa.

(R. Yochanan): Hefker in front of three is Hefker. In front of two, it is not.

(R. Yehoshua ben Levi): Mid'Oraisa, even in front of one person it is Hefker. Chachamim required three, in order that one could acquire it and the other two could testify to this.

Shabbos 18a: Beis Shamai require Shevisas Kelim (allowing one's vessels to rest on Shabbos).

Question: Why do Beis Shamai permit (to leave from before Shabbos) beer brewing in a barrel, oil burning in a lamp, food in a pot on a fire, and the Korban Pesach on a spit?

Answer: Beis Shamai hold that he makes these Kelim Hefker.

Bava Metzi'a 30b: A man asked R. Yishmael to help him load up wood. R. Yishmael bought the wood (to avoid having to help), and made it Hefker. The man acquired the wood and asked for help. R. Yishmael again bought it and made it Hefker. The man was about to acquire it again.

R. Yishmael: It is Hefker to all except for you.

Really, Hefker must be for rich and poor, like Shemitah. R. Yishmael made it Hefker for all. He deceived the man so that he would not take it again.

Rishonim

Rif and R. Shimshon (brought in Rosh 4:11): R. Yochanan taught that Hefker is valid in front of three, but not in front of two. We require three in order that one could acquire it in front of two witnesses.

Rambam (Hilchos Nedarim 2:16): Mid'Oraisa Hefker works even if in front of one, and the land is exempt from Ma'aser. Mid'Rabanan, it is not Hefker unless it was in front of three, so that one could acquire it in front of two witnesses.

R. Shimshon (ibid.): R. Yehoshua ben Levi taught that mid'Oraisa, even in front of one person it is Hefker. The Halachah always follows him against R. Yochanan. Mid'Oraisa it is Hefker even (when he is) by himself. Therefore, one can make his barrel, spit, lamp and pot Hefker (to avoid Shevisas Kelim) by himself. If one lent his animal to a Nochri and it was not returned before Shabbos, he can make it Hefker by himself.

Question (Rosh): R. Yehoshua said that mid'Oraisa, Hefker is valid in front of one. What is the source to say that it is valid by himself? It is logical to distinguish these. R. Yehoshua does not require witnesses, but he requires one person who could take it out of the Reshus of the Mafkir. This person would be believed to say that it was made Hefker, Migo (since) he could say 'you gave it to me for a gift.' It is good (not to rely on R. Shimshon, rather) to make it Hefker in front of one's wife or a member of his household.

Answer (Rosh): Perhaps R. Shimshon learns from Shemitah, like the Yerushalmi (Shevi'is 5:1). This refers to Hefker of land, like Shevi'is. We apply it to exempt from Ma'aser. R. Yochanan requires three, for (one cannot take land, so) without three one cannot acquire it and keep it (if the Mafkir will deny that he made it Hefker). R. Yehoshua does not require three, for he does not require that it can leave his Reshus. Therefore, Hefker works even by himself. 'In front of one' is not precise.

Tosfos (Nedarim 46a DH R. Yehoshua, citing R. Tam): R. Yochanan holds like R. Yosi, and R. Yehoshua ben Levi holds like Chachamim. R. Yehoshua requires three only to exempt from Ma'aser.

Rosh (Shabbos 1:33): Hefker must be in front of three. This is mid'Rabanan, in order to publicize the matter. Mid'Oraisa it is Hefker even if he was alone. Some are lenient to rent and lend their animals to Nochrim; they make the animals Hefker (to avoid transgressing Shevisas Behemto). This is improper. Chachamim allowed (Hefker not in front of three) only when there is no other solution (i.e. according to Beis Shamai, beer in a barrel...) Also, the rental is known, but the Hefker is not (so it looks like the Yisrael transgresses).

Beis Yosef (OC 246 DH u'Mah she'Chosav Afilu): It seems that this is when he is Mafkir by himself, like Hefker of a barrel... If he is Mafkir in front of three, this becomes known, so perhaps he may rent to a Nochri l'Chatchilah. Perhaps this is even if he is Mafkir in front of one. The Tur forbids when he is Mafkir by himself. The Mordechai says that b'Di'eved, if one lent to a Nochri he must be Mafkir it in front of three. Perhaps it is actually l'Chatchilah in front of three; the Mordechai discusses a typical case. Normally, one would not be Mafkir in front of three in order to lend to a Nochri.

Rashba (Shabbos 18b DH Gigis): Normally, Hefker requires three. Here it does not, for presumably he has full intent to be Mafkir, to evade the Isur. It seems to me that Chachamim stipulate to make the Kelim Hefker. Not everyone knows that this is necessary!

Nimukei Yosef (Bava Metzi'a 16b DH Hadar): Presumably, R. Yishmael was alone with the man. If not, the man would have asked someone else to help him load! Even though Hefker requires three mid'Rabanan, R. Yishmael was not concerned lest someone transgress theft mid'Rabanan (by taking it). Others say that we require three only to exempt from Ma'aser. An Aveidah mi'Da'as (one who leaves something where it will get ruined) is Hefker, even if no one else was around!

Poskim

Shulchan Aruch (CM 273:7,9): Mid'Oraisa, Hefker works even if he was in front of one, and the land is exempt from Ma'aser. Mid'Rabanan, it is not Hefker unless it was in front of three, so that one could acquire it in front of two witnesses.

Rema: Some say that it is Hefker even if it was not in front of anyone else.

Gra (8): This opinion holds that mid'Oraisa, Hefker need not be in front of anyone else, and that mid'Rabanan, it suffices to be in front of one other. Three are needed only to exempt from Ma'aser. We learn from Shabbos 18b and Bava Metzi'a 30b.

Shulchan Aruch (OC 246:3): If one lent his animal to a Nochri and it will not returned before Shabbos, before Shabbos he should make it Hefker when he is alone, or say 'my animal is acquired to the Nochri', to save himself from an Isur mid'Oraisa.

Gra (10): Perush ha'Rosh (Nedarim Sof 45a) says that b'Di'eved it is Hefker even by himself. Tosfos (Shabbos 18b DH Makera) agrees that this helps when there is no alternative. The Mordechai (Shabbos 240) and other Poskim say that Hefker by oneself helps only for Kelim, which he is Mafkir every Shabbos. For an animal, even b'Di'eved three are required.

Rema: If he wants, he can be Mafkir it in front of three people, like other Hefker. No one can acquire it, for surely he only wants to avoid the Isur Shabbos.

Mishnah Berurah (15): L'Chatchilah he should be Mafkir in front of three to fulfill all opinions. In any case he must be Mafkir in front of at least one, even a member of his household, for some require one mid'Oraisa.

Kaf ha'Chayim (43): Since the reason for three is to enable someone to acquire with witnesses, the three should include two Kosher witnesses who are not relatives.