Gemara

(Rav Yehudah): If a Mavoy is 15 Amos wide, one takes a plank three Amos wide and puts it two Amos from a wall.

Question: Why don't we erect three planks, each one Amah wide, with an Amah between planks?

Answer #1: We must say that Omed k'Parutz (Mechitzos that are as wide as the space between them) does not permit!

Rejection (and Answer #2): Really, Omed k'Parutz permits. Here is different, because the air from both sides [of the plank in the middle] is Mevatel it.

(Beraisa): Hide of a toilet seat and its hole join to the Shi'ur of a Tefach [to be an Ohel].

(Rav Dimi): The hole can be [up to] two fingers, with two fingers [of hide] on each side.

(Ravin): It can be [up to] one finger, with one and a half fingers on each side.

Rav Dimi: We argue about Parutz k'Omed (there is as much air as solid wall. I permit, and he forbids.)

16b (Mishnah): A caravan that camped in a valley may surround it with three ropes, one above the other, as long as there are not three Tefachim between successive ropes. The total thickness of the ropes must be more than a Tefach, so that the top of the top rope will be at least 10 Tefachim.

(Abaye): If Omed Merubah Al ha'Parutz permitted a horizontal Mechitzah, the thickness could be any amount, the spaces between each rope and another [or the ground] would be [just less than] three, three and four Tefachim!

We cannot say that the top gap is four, for the air above and below the top rope would be Mevatel it.

Rishonim

Rashba (10b Sham'at) and Ritva (DH umid'Hadrinan): We can say that Omed Merubah Al ha'Parutz permits when the air on both sides is not Mevatel, e.g. one plank is on the right side of the Mavoy, one is on the left, and one is in between. Alternatively, it applies in other cases such as a Sukah that has spits or bed boards (Pasul Schach) and Kosher Schach. There is no air to be Mevatel the Kosher Schach.

Ritva (ibid.): Sukah 15a says that the argument of Omed Merubah Al ha'Parutz applies to Pasul and Kosher Schach. One might have thought here is different, because each airspace is bigger than the wall in between. If the wall were as big as the space, it would not be Batel. This is why we suggested making a plank an Amah wide, for then it is equal to the space on one side.

Tosfos (10b DH v'Etzbva'ayim): The hide of a toilet seat is a Kli. 'The air on both sides is Mevatel' does not apply to it. It applies only to fixed Mechitzos, like above, and ropes around an encampment. The Ritzva says that 'the air on both sides is Mevatel' does not apply to regular air. It is only when we consider it sealed, like above. Regarding the ropes, the air above 10 is considered sealed. Air that is not part of the Mechitzah is not Mevatel. Above (5a), R. Ami and R. Asi say that a plank four wide permits a breach up to 10. We do not say that the air on this side and of the world (i.e. outside) is Mevatel. If we would distinguish the beginning of a Mavoy from the end [this would not be a proof].

Rashba (10b DH v'Etzba'ayim): 'The air on both sides is Mevatel' does not apply here (a toilet seat), for the reasons I wrote above. Also, it is a Kli, and 'the air on both sides is Mevatel' does not apply a Kli.

Maharil (Teshuvos ha'Chadashos 4:2): My Rebbi was particular to make the shoulders of the Talis Katan wide, lest the air on both sides be Mevatel. He did not rely on a majority of wall from both sides [of the hole in the middle], nor on [the leniency of] a hole destined to be filled.

Poskim

Shulchan Aruch (OC 16:1): The Shi'ur to obligate putting Tzitzis on a Talis is long and wide enough to cover Rosho v'Rubo of a child old enough to go in the market by himself.

Magen Avraham (Reish ha'Siman): One should make the shoulders (on the side of the neck-opening) wide, lest they be Batel to the air on both sides of them.

Machatzis ha'Shekel: This is a borrowed expression. (The concern here is not the same as regarding a Mavoy.) Really, the concern is that the shoulders be considered a [part of the] garment, and not mere straps. 'The air on both sides is Mevatel' does not apply, like Tosfos said. The Magen Avraham (362:16) says 'normally, air is not Mevatel. We say so only about air that we consider sealed.'

Question: (One may not carry in a Karfef (an enclosed area without a roof) more than Beis Sa'atayim if it was not surrounded for Dirah, e.g. it is seeded.) If most of a garden is seeded in patches, do we consider each patch by itself, and it is Batel to the fallow patches? Or, since the majority is seeded, it is forbidden? If it is forbidden, when the minority is seeded, and it is more than Sa'atayim, so it is like Karmelis, does it forbid carrying even where it is not seeded?

Answer (Chacham Tzvi 59): I forbid the first case, but permit the latter. We never find that in two places, neither of which has a Shi'ur, and there is Heter between them, that they join to forbid, if not due to the majority, which always joins to be Mevatel the minority. Air and Pasul Schach are Pasul mid'Oraisa, yet they do not join if there is Heter in between, even if the Heter is less than three Tefachim. I am unsure when there is less than three. I am not concerned for the Rema (YD 371:4), who says that we do not say Lavud to be stringent, for he overlooked an explicit Gemara (Eruvin 25a). Rather, perhaps the fallow land in between is like Kosher Schach in between Pasul Schach. The two Pesulim do not join even if there is less than three Tefachim of Kosher Schach in between. Lavud applies only to air, but not to anything with substance.

She'alas Ya'avetz (1:20, cited in Sha'arei Teshuvah 16:1): The shoulders of a Talis Katan need not be wide. We are not concerned lest the air on both sides is Mevatel, like the Acharonim bring from Maharil. They overlooked Tosfos, who applies 'the air on both sides is Mevatel' only to fixed Mechitzos, but not to a Kli. My father said so from reasoning, that 'the air on both sides is Mevatel' is only when we need to consider air to be part of a Mechitzah through Lavud. A small amount of solid wall is Batel in a majority of air. (I later saw that he explicitly said so in Teshuvah 59.) Here, we need not join air to the garment, so air is not Mevatel [the shoulder pieces]. Tosfos says like this.

Tosefes Shabbos (362:23, cited in R. Akiva Eiger): The Magen Avraham brings from Tosfos that 'the air on both sides is Mevatel' applies only to air that pertains to Mechitzos, and we consider it sealed, e.g. the air above the top rope until the sky, for the air of Reshus ha'Yachid goes up to the sky. The same applies when a Mavoy has a breach above 10 Tefachim (perhaps this is a printing mistake, and it should say 10 Amos - PF), and one erected a stick in the middle. Then we say the air is Mevatel. We do not say so about Stam air that is not part of the Mechitzah. In Siman 16, the Magen Avraham brought from Maharil that one should make the shoulder pieces wide, lest they be Batel to the air on both sides of them. It seems that he argues with Tosfos.

Eshel Avraham (Butshats, DH Udos): What is the limit for which we are concerned for Bitul? I thought to say a Tefach on each side, but I did not see anyone give a Shi'ur. It seems that we require Omed Merubah Al ha'Parutz. Maharil need not argue with Tosfos. Presumably Maharil discusses a minimal size Talis Katan of three quarters of an Amah long and half an Amah wide. The neck opening is needed for the Shi'ur. Tosfos agrees that 'the air on both sides is Mevatel' applies in such a case. Perhaps it helps for the [sum width] shoulder pieces to be slightly more than the opening, like we say regarding Shabbos that 'the air on both sides is Mevatel' does not apply when Omed Merubah Al ha'Parutz. This requires investigation. If each shoulder piece is bigger than the hole, all agree that there is no concern. This is the custom for our Talesim Ketanim. Below (towards the stomach) no fabric is missing, just a slit allows the sides to separate. There is no concern for this at all, especially since Maharil agrees with Tosfos, and he is stringent only when the air is needed for the Shi'ur.

Eshel Avraham: However, perhaps Maharil discusses even a Talis Katan bigger than the minimal Shi'ur. Even so, he does not argue with Tosfos. The air is Mevatel the shoulder pieces. It is as if the Talis was cut into two. The front and back are separate. Each has only two Tzitzis. If so, always the air joins [for Bitul]. However, this is wrong. Only air that must join [for a Shi'ur] is Mevatel. The shoulder pieces are intact. There is one connected garment with four corners. There is no concern for a gap in a garment. If someone wrote the Shi'ur of a Tefach, it is for a minimal size garment half an Amah (three Tefachim) wide. If each shoulder pieces is a Tefach, this is easily the majority. Any amount above three fingers is enough [to make a majority]. Even if the garment is bigger, it suffices to have this amount, which suffices for a minimal size garment.