Sukkah, Chapter 1, Mishnah 3. In the previous mishnah we dealt with the matter of partitions between a person sitting in the sukkah and the sechach, and our mishnah continues with that same subject, discussing a sheet spread out in the sukkah.
Three ways in which a sheet invalidates:
A sheet spread over the sechach "mipnei hachamah" - in order to shield those sitting in the sukkah from the sun.
A sheet spread beneath the sechach "mipnei haneshirah" - because leaves falling from the sechach disturb those sitting there.
A sheet spread over a bed that has four posts - which forms an independent tent in its own right.
In each of these three cases the mishnah rules: "pesulah" - it is invalid, and anyone sitting beneath it has not fulfilled the mitzvah. The reason is the same in all of them: the sheet, whether it rests on top of the sechach, whether it rests beneath it, or whether it is part of an independent structure, functions as a tent in its own right, and it forms a partition between the person sitting there and the sechach above him.
Spreading a sheet over the bedposts:
One is, however, permitted to spread the sheet over the "naklitei hamittah" - the bedposts. What are naklitei hamittah? A single beam standing in the middle at the head of the bed, and a single beam standing at the foot of the bed, with the sheet spread over them, creating a slanted structure - a kind of triangular tent that rises and converges to a single point.
The minimum measure of a tent, for it to be considered a roof that interposes, is a width of at least a tefach. In this structure, since the sheet converges at the top to a point and has no area a tefach wide, it is not considered a tent, and it does not form a partition between the one sitting there and the sechach - and therefore it is permitted to sit beneath it.
In summary: Our mishnah continues the subject of a partition between the one sitting in the sukkah and the sechach. A sheet spread above the sechach to shield from the sun, or beneath it because of falling leaves, or over a bed that has four posts, is considered a tent in its own right and forms a partition, and is therefore invalid. In contrast, a sheet spread over the bedposts, which converges to a point and has no width of a tefach, is not considered a tent and does not form a partition, and it is permitted to sit beneath it.