Edit: The theorem that was proved is here: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szemer%C3%A9di_theorem , basically a statement that either there can be many unique elements in a sum matrix or many unique elements in a product matrix, but not both at the same time. They proved that it can be shown as $ck^{1+\epsilon}$, and the conjecture was that you can take the $\epsilon$ arbitrarily close to 1, though current results have it near 1/3.
Regarding the Erdos and Szemeredi sum/product conjecture about the unique values in sum and product matrices, I was looking at the special case of 1, 2, ... n, the simplest arithmetic progression. It's trivial to show that for the set containing the first n positive integers, in the sum matrix there are $\frac{n^2+n}{2}$ non-trivial values, and 2n-1 of them are unique.
In the corresponding product matrix, it's slightly more complicated. As each new integer n is added to the set, n non-trivial values are added, but many of them can and will be duplicates. Define a function $\alpha(n)$ that returns the smallest prime factor of n, which is defined as 1 for n = 1. Then the number of duplicates added for integer n is $\beta(n) = \frac{n}{\alpha(n)} - 1$. It's clear to see that for n prime where $\alpha(n) = n$, zero duplicates are added, and in general numbers with low smallest prime factors like 2 and 3 add more duplicates and numbers with higher smallest prime factors add less duplicates.
If we look at the asymptotic behavior, according to Average smallest prime factors, $\alpha(n) \sim \frac{n}{2\log{n}}$, therefore $\beta(n) \sim 2\log{n}-1$. Then we have unique entries in the product matrix (asymptotically): $$\sum_{k=1}^{n}k - 2\log{k}+1$$ $$= \frac{n^2+n}{2} + n -2\sum_{k=1}^{n}\log{k}$$ $$= \frac{n^2+3n}{2} - 2\log{n!}$$ $$\sim \frac{n^2+3n}{2} -2n\log{n}$$ and finally, if you add the unique entries from the sum matrix, you have: $$\sim \frac{n^2-4n\log{n}+7n-2}{2}$$
This appears to line up with the Erdos and Szemeredi conjecture of $n^{2-\epsilon}$ for any $\epsilon > 0$. Presumably you could then show similar results for the special case smallest geometric progression, and show the results hold for all progressions, not just the basic cases. Is that enough to actually prove the conjecture, or are there non-trivial cases that aren't arithmetic or geometric progressions that you would have to prove are always greater than this lower bound for the special cases?