On product Schur triples in the integers

Schur’s theorem states that in any k-coloring of the set of integers [n] there is a monochromatic solution to a + b = c , provided n is sufficiently large. Abbott and Wang studied the size of the largest subset of [n] such that there is a k-coloring avoiding a monochromatic a + b = c. This led to the exploration of related problems, such as the minimum number of monochromatic a + b = c in k-colorings of [n] and the probability threshold for a random subset of [n] to have a monochromatic  a + b = c in any k-coloring. In this paper, we study natural generalizations of these problems to products  ab = c, in deterministic, random, and randomly perturbed environments.