If two people do melocho together, each doing one part, they are exempt. An example would be זה עוקר וזה מניח- one does an akira and the other does the hanacha.

If two people both carry together, R Meir says they are both liable. R shimon says they are both exempt. R Yehuda says it depends- if they are both capable of doing it themselves they are exempt. But if they’re Incapable of doing it themselves they are liable.

If reuven and shimon together carried an item- reuven could do it alone and shimon could not, then shimon’s contribution is merely “insignificant assistance” and everyone agrees that reuven is liable and not shimon.

This concept of insignificant assistance is relevant to other areas of Halacha as well. Examples include a Kohen standing on the floor of the Temple but leaning somewhat on a vessel or rock, or a Kohen doing service with the right hand but getting some assistance from the left hand. Additionally, the law regarding zav is that there’s tumas midras when the majority of zavs weight is supported. If a zav rides an animal and there are shawls under the legs of the animal each of the legs are Tahor bc one leg relative to the other three is insignificant support. (Animal cns stand on three legs) In contrast, if a zav is sitting on four legged couch with Shawls underneath, all the shawls are tamei bc they are all needed for the couch to stay.

In a scenario when two people performing melocho together are liable, the Gemara wonders if one shiur/measurement suffices time obligate them both or whether there must be two shiurim. The gemara favors that one size obligates Both.

If beis din makes mistake in Halacha ruling and someone sins bc of their mistake, there’s a tanaaic dispute if they are exempts