Shulchan Aruch Chapter 632:  Things That are Invalid For The Sechach (4 Paragraphs)
Note: REMA in brackets

1.  Invaid Sechach is not valid if it is 4 handbreaths or greater in size.  However, less than 4 handbreaths is valid and it is permitted to sleep under it {1}.  This applies if the invalid Sechach is over 4 cubits from the wall.  However, if less than 4 cubits are between the invalid Sechach and the wall, it is valid (with over 4 handbreaths of invalid Sechach) because we say that "the wall is curved" or since the invalid Sechach touches the wall it looks like it is part of the wall and the wall has an angle to it (so long as enough valid Sechach is there to make the Succah valid){2}.  This is the reason why if one opens a hole in the roof of one's home and puts Sechach on it it is valid if the remaining home roof is within 4 cubits of the walls.  However, one may not sleep under the portion which has invalid Sechach of greater than 4x4 handbreaths.  What I previously wrote only applies when one has a large Sucah where the Kosher Sechach predominates over the invalid Sechach by being 7x7 handbreaths or greater in total size.  However, a small Sucah that is only 7x7 handbreaths in size, greater than 3 handbreaths invalidate it whether in the middle of the Sechach or from the sides.  Less than 3 handbreaths are valid and one can sleep under it.  One may build on to the invalid Succah to make it Kosher.

2.  Air is invalid with 3 handbreaths.  It makes no difference if the Succah is big or small or if the portion is in the middle of the Sechach or from the sides.  One can add Sechach to make the Succah valid.  However, one may not sleep under the air.  [This only applies if the air is prevalent all over the Succah or if there is not enough of valid area to fit one's head and most of one's body under.  But otherwize, the Succah is valid to sleep under since there is no Succah without small air spaces.  Where he writes that the Succah is invalid with over 3 handbreaths of air and 4 handbreaths of invalid Sechach is only referring to where it divides the Succah in two and there does not remain enough of the Sechach to be valid in one area.  However, if a Kosher area of Sechach remains then that area is valid.  This is even true outside the Succah if it is attached with walls.]

3.  Sechach that is invalid if greater than 4 handbreaths and air that is invalid if greater than 3 handbreaths do not combine to invalidate.  Therefore, if there are exactly 3 handbreaths of air in one place with a little invalid Sechach then the Succah is valid.  This only applies to a large Succah.  However, a small Succah with only a size of 7x7 handbreaths, they do combine together to invalidate by only 3x3 handbreaths.

4.  If there is one patch of invalid Sechach 2 handbreaths in size plus another patch of invalid Sechach 2 handbreaths in size and a patch of air between them less than 3 handbreaths, then it is in doubt if the two join together to make the Succah invalid.
{1} M.B. One should be strict since there are authorities who say that one may not sleep under an invalid area unless it is less than 3X3 handbreaths.
{2} M.B.  This implies that it would be invalid if the Sechach did not touch the wall. We should be lenient and say that if the invalid part is within 3 handbreaths of the wall, this rule applies.
Translated by Dr. Jay Dinovitser DO
Free for Personal Use Only