A solution to the Erd\H{o}s--S\'{a}rk\"{o}zy product-density problem

Color each positive integer red or blue, and call an integer a \emph{same-color product} if it is the product of two red numbers or of two blue numbers. Does there exist a universal constant $c > 0$ such that: For each fixed coloring, and for all sufficiently large $M$ (where ``sufficiently large'' may depend on the coloring), at least $cM$ of the integers among $1$, $2$, $\ldots$, $M$ are same-colored products (allowing for equal factors, and where each distinct product is counted once)? This problem was given (in an equivalent way) in Guy's famous text on unsolved problems in number theory, and was posed (in equivalent ways) by Erd\H{o}s and S\'{a}rk"{o}zy in 1990, in the proceedings of the 1988 Banff number theory conference, and 1993 [\emph{Colloq.\ Math.}], and this problem seems to have remained open. We succeed in solving this problem, by proving that no such universal constant exists.