Let $A$ be a set of nonzero integers. We say a set $B$ is sumfree if there do not exist $x,y,z\in B$ with $x+y=z$. Erdos proved in 1965 that for any set $A$ of nonzero integers, there exists a sumfree $B\subset A$ such that $|B|>|A|/3$.
Specifically, I am having trouble understanding a detail in a proof of this theorem. The proof I am referring to occurs in the first section of Additive Combinatorics by Tao & Vu (pg. 4) and in section 20.2 of Extremal Combinatorics With Applications In Computer Science by Jukna (pgs. 246-247). Here are snapshots I took of the proofs from each book (Tao & Vu first, then Jukna's proof):
The thing I do not get is the stipulation for the size of $p$. In Jukna's book, it is required that $p>2\max_i|a_i|$ where the $a_i$ are the elements of $A$. In Tao & Vu it is required that $A\subset[-p/3,p/3]$. Firstly, why do we need $p$ to be large at all? Does this affect the probabilities in any way? Secondly, clearly the "largeness" of $p$ varies in the two books and yet still the proofs work, so I'd like to understand what restrictions on the size of $p$ is really essential. These sizes don't seem to be mentioned in the proofs after they are stated, so I know I must be missing something.
Note: I have edited my answer significantly in that I now provide the actual proofs that I am concerned with (before I just mentioned where they are to be found). Hopefully this will give at least one person the motivation (or wherewithal) to respond.
Well, we clearly need $p$ to be large enough that $A$ has no two elements which are congruent modulo $p$: otherwise, the step of viewing $A$ as a subset of $Z_p$ breaks down. As for why $p$ needs to be any larger than the largest element of $A$, my guess here is that this is just to maintain the nice property that $B \subset A$ is sum-free modulo $p$ iff $B$ is sum-free in the ordinary sense. For the purposes of the theorem, though, all that's really needed is the forward implication, which holds even if $p$ is somewhat smaller.