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1)

(a) A Keren Zavis ha'Semuchah li'Reshus ha'Rabim' speaks either when someone's house protrudes into a corner in the street, and the owner donated a little bit of his land to the public. The result is that the house obstructs the people from walking straight along the street, and they are forced to make a slight detour, giving that part of the street the Din of a Karmelis.
Alternatively, we are speaking about a house which is built at an angle, so that one corner of the house lies right next to the street, in which case, it becomes difficult for people to walk straight past at that point (because they will brush against the corner of the house.

(b) 'Between the pillars' is an area in the Reshus ha'Rabim full of pillars. The pillars are placed in unaligned rows, and are designated for merchants to hang their wares whilst engaged in business. The unaligned rows of pillars make it difficult for the public to walk straight through, which is why it has the Din of a Karmelis.

(c) According to Rav Yehudah, the area of 'Bein ha'Amudim' is considered a Reshus ha'Rabim, either because that area is readily accessible to the public (in spite of the fact that they cannot easily walk straight through), or because sometimes, when the street is full, the public do use that area as a thoroughfare.
The Itztava, on the other hand, which consists of benches, renders the area totally inaccessible to the public, and is therefore only a Karmelis.

(a) The Gemara concludes, that anything within three Tefachim of the Reshus ha'Rabim (even excrement) is Bateil to the Reshus ha'Rabim. Consequently, one would be Chayav for throwing an article, if it landed on anything that is less than three Tefachim tall.

(b) If the object lands on top of something that is between three and ten Tefachim tall, then the thrower is Patur, because the top of a brick is a Makom Petur (provided it is less than four by four Tefachim). Whereas if it were to stick to the side of the brick, he would be Chayav, since the air of the Reshus ha'Rabim is part of the Reshus ha'Rabim.

(c) We have learnt that a house whose walls are less than ten Tefachim on the inside has the Din of a Karmelis, and if the roof makes up the ten Tefachim, one may carry on the roof, because the roof of the house has the Din of a Reshus ha'Yachid. So we see that it is *without* walls of ten Tefachim that a Reshus becomes a Karmelis, not *with* them.

(d) This contention applied to a Bik'ah that was surrounded by a wall, but was not initially designated for habitation purposes. Halachically, this is a Reshus ha'Yachid, which the Rabbanan gave the Din of a Karmelis. According to the contention of the Gemara, we thought that it is a Karmelis d'Oraysa.

(a) What Rav Sheishes meant was that a Karmelis only extends to the height of ten Tefachim, but no higher.

(b) He could not have been referring to a Reshus ha'Yachid, because Rav Chisda has taught us that a Reshus ha'Yachid extends even up to he sky.

(c) Nor can he have meant that a Reshus ha'Rabim does not extend beyond ten Tefachim, because we know already from a Mishnah in 'ha'Zoreik' that even the air of a Reshus ha'Rabim above ten Tefachim has the Din of a Makom Petur.

(d) As we just write, the air above ten Tefachim in a Reshus ha'Rabim, is considered a Makom Petur.

4)
1. Chazal gave a Karmelis the leniency of a Reshus ha'Yachid, inasmuch as if it comprises an area of less than four by four Tefachim, it has the Din of a Makom Petur, and one is permitted to carry from either of the two major Reshuyos on to it or vice-versa.
2. And they also gave it the leniency of a Reshus ha'Rabim, inasmuch as, above ten Tefachim, it no longer has the Din of a Reshus ha'Rabim, but of a Makom Petur.

5)

(a) If a house was ten Tefachim tall from the outside, but not from the inside, one would be able to dig a hole in the ground measuring four by four Tefachim, and to carry there, even outside the hole. This is because it has the Din of 'Chorei Reshus ha'Yachid' (and everyone agrees that, 'Chorei Reshus ha'Yachid, ki'Reshus ha'Yachid').
If the hole would be within three Tefachim of the walls of the house, we would not need to come on to the Din of 'Chorei Reshus ha'Yachid', since it would then be an integral part of the Reshus ha'Yachid (See Tosfos Yeshanim, Tosfos d.h. 've'Im').

(b) The reason that a 'Keren Zavis' is considered an independent Reshus, is because the people in the street are not easily able to use it as they walk along, unlike holes in the wall, which they *are*.

(c) If a Keren Zavis is more than four by four Tefachim, it will be a Karmelis, if and if not, a Makom Petur.

(a) First of all, an object that is thrown more than four Amos, and lands in a cavity in a wall, usually bounces out again. And secondly, it is evident from the Seifa, that the Mishnah is speaking about an object which actually sticks to the wall, and not one which falls into a cavity. Why is that?
Because the Mishnah says that if the article lands above ten Tefachim, it is as if it landed in the air (as if it has not landed on a Makom Chashuv). Whereas if it lands in a cavity, then it has landed on a Makom Chashuv!

(b) Nor can we establish that it fell in a cavity of less than four by four Tefachim (where it is still appropriate to call it 'in the air', because we are dealing with a Stam Mishnah, whose author is Rebbi Meir, and Rebbi Meir holds that any area of more than four than four Tefachim, which is partially carved out, we consider it as if it had been completely carved. Consequently, according to him, if the article lands in a small cavity, on an area which is larger than four by four Tefachim, he will be Chayav. We are therefore forced to say that above ten Tefachim, is considered as if the article had landed in the air, speaks when there was no cavity at all - so it must be something like a fig, which stuck to the wall above ten Tefachim.

(a) We learn from Rav Chisda, who says that someone is Chayav on Shabbos for throwing from a Reshus ha'Rabim on to a cane a hundred Amos tall in the Reshus ha'Yachid, that the Reshus ha'Yachid extends to the sky.

(b) The Gemara initially thinks that the Rabbanan say Patur because the Ziz (which usually means a ledge or a plank) is not four by four Tefachim on top. Consequently, Rebbi, who holds that he is Chayav, does not require four by four in a Reshus ha'Yachid - and we will have to say that Rav Chisda holds like Rebbi.

(c) We could establish the Machlokes by a tree in a Reshus ha'Yachid , whose branches stretch across into the street. In reality, the Rabbanan also agree with Rav Chisda that one is Chayav, even when the object lands on an area of less than four by four Tefachim (See Tosfos, d.h. 'Amar', that Rav Chisda's Din is confined to a Reshus ha'Yachid). The Rabbanan's reason for saying Patur is, because, according to them, the branch goes after the trunk, so that they consider the branch to be in the Reshus ha'Yachid, where the trunk is.

(d) According to Rashi's first explanation, the tree is standing entirely in the Reshus ha'Rabim. Both Tena'im agree that one requires four by four Tefachim, and the Rabbanan's reason is because, in fact, the branch of the tree (on which the object landed) *does* have an area of four by four Tefachim. How is that?
Because they maintain that the branch goes after the tree (not to establish the *location* of the branch, but to establish its *area*). It therefore transpires, that both Rebbi and the Chachamim require four Tefachim by four Tefachim - at least, in the Reshus ha'Rabim, and consequently, Rav Chisda is not saying like either of them (though he does not argue with them either, because *he* is speaking in the Reshus ha'Yachid, whilst *they* are speaking in the Reshus ha'Rabim).

(e) According to Rashi's second explanation, Rebbi and the Rabbanan argue about whether the branch follows the *location* of the tree (not the area). Consequently, both Rebbi and the Rabbanan hold like Rav Chisda, because they both agree, that, if the branch would follow the tree (location-wise), he would be Chayav, because, in a Reshus ha'Yachid, one does not require four Tefachim.