Motivated by applications of Ramsey theory, Schur's theorem states that for any partition of the positive integers into a finite number of parts, one of the parts contains three integers $x, y, z$ with $x+y = z$.
I consider 2-coloring of the first $m$ integers. I suspect that there are more distinct product triples ($z_1=x*y$) than distinct sum triples ($z_2=x+y$). I am interested in finding bounds on the number of distinct sum triples vs distinct product triples in any 2-coloring of first $m$ integers as a function of $m$.
The lower bound is the maximum number of distinct sum triples (product triples ) that exist in some color class over all possible 2-colorings . The 2-coloring is balanced (the cardinally of each color class is greater than $\epsilon *m$ for some constant $\epsilon \gt 0$).
What is known about the asymptotics of distinct sum triples vs distinct product triples?