Why is the inner product of -1,+1 binary variables at most $n-2$ and not at most $n-1$?
I just noticed that if we have one disagreement then we change one single entry from +1 to -1. Thus, the summation decrease by 1 because we are missing one of the +1's in the summation but on top of that because they are Rademacher vectors instead of simple binary 0,1 then it means that the +1 actually changes to -1, meaning the inner product decreases by an additional 1. 

So it decreases by 2 cuz:


we lose a +1 in the summation
but we also replace the +1 with a -1


so total decrease is by two leading to $ \leq n-2$