In his paper "A survey of problems in combinatorial number theory", on page 110, Erdos writes:
Graham conjectured: Let $1 \le a_1 < a_2 < \cdots < a_n$ be $n$ integers. Then $$ \max_{i,j} a_j/(a_i, a_j) \ge n. $$ Szemeredi proved this recently. The proof is not yet published.
I tried to find the proof among Szemeredi's publications, but failed. Can anyone provide me with any reference (or maybe the proof itself) to this theorem?
You might be interested in this paper : "On a conjecture of R. L. Graham" by R. Balasubramanian and K. Soundararajan. They prove Graham's conjecture with the additional condition $(a_1,a_2,\cdots,a_n)=1$ a condition which can be obtained without the loss of generality. (So yes the link is indeed the proof of the conjecture.)
In the introduction, the authors say that Szemeredi gave a proof for $n=p$. So it may be the case in which there was some communication error(?) between Szemeredi and Erdos.
Cheers! :-)