Gemara

(Mishnah): If the Oni put his hand inside... [and put an object in the hand of the Ba'al ha'Bayis, he is Chayav].

Question: Why is he Chayav? Akirah and Hanachah must be from [and on] a place that is four by four [Tefachim]!

5a - Answer (R. Avahu): The case is, the receiving hand was within three Tefachim of the ground. (It is Batul to the ground, so it is considered to be four by four.)

7a (Rabah bar Shila): If one threw something [four Amos] in Reshus ha'Rabim and it landed on top of a brick, he is exempt.

(Abaye and Rava): The brick must be [at least] three Tefachim tall. If not, many people trample on it.

Regarding prickly bushes, even if they are not three Tefachim [they are not Reshus ha'Rabim, for people do not step on them].

(Chiya bar Rav): Prickly bushes are Reshus ha'Rabim [people wearing shoes step on them], but excrement is not.

(Rav Ashi): Even excrement is Reshus ha'Rabim.

(Rav Dimi): Karmelis does not apply to an area less than four [by four Tefachim].

(Rav Sheshes): Karmelis applies up to 10 [Tefachim above the ground].

I.e. until 10 Tefachim off the ground is Karmelis, but above is not (it is Mekom Petur), like Shmuel taught.

(Rav Chisda): If one erects a beam in Reshus ha'Yachid with a basket on top, even if it is 100 Amos tall, one who throws something into it from Reshus ha'Rabim is liable, for Reshus ha'Yachid extends to the sky.

8a (Ula): If a pillar in Reshus ha'Rabim is nine Tefachim tall and many people are Mekatef on it (use it to adjust loads), if one threw something and it landed on the pillar, he is liable.

Something [in Reshus ha'Rabim] less than three Tefachim tall [is Reshus ha'Rabim, for] people walk over it;

Something at least three but less than nine, people neither walk over it nor are Mekatef on it;

Something nine tall, many use it to be Mekatef on it.

97a - Question (Rav Chisda or Rav Hamnuna): What is the source that anything within three Tefachim is Lavud (Batul to the ground)?

Answer #1 (The other of Rav Chisda and Rav Hamnuna): Reshus ha'Rabim cannot be paved perfectly flat.

Question: If so, even something three Tefachim should be Lavud!

Answer #2: Rather, it is a tradition from Moshe from Sinai.

100a (Rava): If one threw within three Tefachim on the ground he is exempt unless it rested on top of something.

Eruvin 33a: The Mishnah discusses a tree on which many people are Mekatef, like Ula taught;

(Ula): If a pillar in Reshus ha'Rabim is nine Tefachim tall and many people are Mekatef on it, if one threw something and it landed on the pillar, he is liable.

Rishonim

Rambam (Hilchos Shabbos 14:8): If something in Reshus ha'Rabim is exactly nine Tefachim tall, no more and no less, it is like Reshus ha'Rabim. Its length and width do not matter. Whether it is wide or short, people use it to be Mekatef. If it was more or less than nine, if it is at least four by four it is Karmelis. If it is not four by four, it is Mekom Petur.

Magid Mishneh: The Ra'avad wrote in his Perush that this is only if it is four wide, that people are Mekatef on it, but not if people are not Mekatef on it. The Rashba says that it is proper to be concerned for the Rambam's opinion.

Rambam (10): Anything less than three Tefachim (in Reshus ha'Rabim) is like the ground.

Rosh (Shabbos 1:15): From nine Tefachim until 10, a beam (in Reshus ha'Rabim) is useful to be Mekatef on it, so it is Reshus ha'Rabim. More than 10 it is not, for it became a different Reshus. If it is four wide, it is Reshus ha'Yachid. If not, it is Mekom Petur.

Poskim

Shulchan Aruch (OC 345:10): Anything in Reshus ha'Rabim, if it is not three Tefachim tall, even if it is thorns or excrement, which not many people walk over, it is considered like the ground and it is Reshus ha'Rabim.

Beis Yosef (DH u'Mah she'Chasav Afilu): One might have thought that since people do not walk on thorns or excrement, even if it is less than three Tefachim tall, it is not considered like the ground. The Tur teaches that this is not so. The Gemara gave a general rule that anything less than three Tefachim is Batel to the ground. Rav Ashi taught that even excrement less than three Tefachim tall is like Reshus ha'Rabim.

Kaf ha'Chayim (51): The Tefachim are Sochakos (with spaces between the fingers) or Atzuvos (without spaces), whichever is a stringency.

Shulchan Aruch (ibid.): If it is at least three Tefachim tall but less than nine Tefachim, if it is four by four wide, it is a Karmelis. If it is not four by four, it is Makom Patur.

Beis Yosef (DH v'Im): Rav Dimi taught that if there is no Karmelis less than four by four. Rashi explains that if it is not Karmelis, it is Mekom Petur.

Mishnah Berurah (29): People cannot trample on something more than three Tefachim tall, so it is not Reshus ha'Rabim, but it is not Reshus ha'Yachid unless it is 10 Tefachim tall. Chachamim made it a Karmelis. One may not take from it to a Reshus ha'Rabim or Reshus ha'Yachid.

Mishnah Berurah (30): Even if it is longer than four, we do not join this to its width. It is its own Reshus only if it is four long and four wide.

Shulchan Aruch (ibid.): If it is exactly nine Tefachim tall, and many are Mekatef on it, it is Reshus ha'Rabim, even if it is not four wide. Some say that the same applies if it is between nine Tefachim and 10 Tefachim and many are Mekatef on it. It is Reshus ha'Rabim even if it is not four wide.

Beis Yosef (DH umi'Tes): Rashi explains that since many need it, it is Reshus ha'Rabim, whether it is wide or narrow. The Rambam agrees.

Beis Yosef (DH u'Mah she'Chasav Rabeinu): The Rosh explains that Ula did not say nine to exclude more than nine. He excludes only less than nine. This is reasonable, for even if it is 10 it is proper to be Mekatef on it. Also, Ula gave a reason to exclude less than nine, but not to exclude more than nine. However, the Rambam says that it must be exactly nine.

Taz (8): This is unlike when it is 10 tall, for then it is used with difficulty. This is not considered a use.

Gra (DH Metzumtzamim): The simple words of the Gemara connote that it must be exactly nine.

Magen Avraham (8): If they are not Mekatef on it, even if it is proper to be Mekatef on it, it is not Reshus ha'Rabim (Beis Yosef citing the Ra'avad).

Gra (DH v'Rabim): Eruvin 33 proves that it is only if many are Mekatef on it. This is like the Ra'avad, unlike the Rambam.

Kaf ha'Chayim (57): Many say that since the Beis Yosef brought the Rashba, who is concerned for the Rambam's opinion, surely he did not rule leniently against the Rambam.

Mishnah Berurah (31): When it is nine tall, it is normal to be Mekatef on it. If it is less than nine, even if many are Mekatef on it, it is not Reshus ha'Rabim.

Gra (DH Afilu): The Gemara mentioned four wide only regarding Reshus ha'Yachid. Also Karmelis is only if it is at least four wide, for it has the leniencies of Reshus ha'Yachid. Four wide was never said regarding Reshus ha'Rabim. (Therefore, a beam to be Mekatef on need not be four wide - Damesek Eliezer.) We require Akirah and Hanachah on a place four by four, both in Reshus ha'Yachid and Reshus ha'Rabim. The Rishonim challenged Rav Chisda's law of a beam in Reshus ha'Yachid (and concluded that no Tana'im require Hanachah on a place four by four in Reshus ha'Yachid - Damesek Eliezer). Also in Reshus ha'Rabim we never require a place four by four. This is not needed if it is within three Tefachim of the ground, like it says on Daf 5a and 100a. If one threw within three of the ground, Rava obligates only if it rested on top of something. Above three is either Karmelis (if it is four by four wide) or Mekom Petur (if it is not). Above 10 is Reshus ha'Yachid.

Damesek Eliezer: When something stuck on the side of something standing in Reshus ha'Rabim, or if one throws into a hole in Reshus ha'Rabim, we require four by four! However, the Rashba holds like the Riva, that on the side does not require four by four. Since it is over the ground, it is considered like the ground.

Gra (ibid.): (Damesek Eliezer - the Rashba and Ran) answered that we discuss (resting on) an animal or person, which is not its own Reshus. Other things do not require four by four. The Ra'avad, Rashba, Ran, Rashi and Magid Mishneh say so. The Ra'avad rejected this and gave another answer, that we discuss a beam nine Tefachim tall. This is because he holds that it must be four by four. We need not give such an answer (of such a limited case), for in Reshus ha'Yachid (Damesek Eliezer - all agree that one is liable when it lands on a thin post, so) one must say that we discuss (i.e. require Akirah and Hanachah on something four by four, regarding) an animal (or person).

Kaf ha'Chayim (58): The Birkei Yosef brings from the Rashba that one should be concerned for the Ra'avad's opinion, that it is Reshus ha'Rabim only if it is four wide. However, the Acharonim hold like the Shulchan Aruch.

Mishnah Berurah (36): We hold like the first opinion, for many argue with the latter opinion.

Shulchan Aruch (ibid.): If it is 10 or more, if it is four wide it is Reshus ha'Yachid. If not, it is Mekom Petur, even if there is room in it to carve out and complete four.

Mishnah Berurah (40): That is to say, even if the post is next to a wall, and if we would consider the wall and post to be one joint entity it would contain within it a square four by four - nevertheless, since the post itself does not contain such a square it is a Mekom Patur.