Monochromatic Translated Product and Answering Sahasrabudhe’s Conjecture

Ramsey theory is an area of combinatorics that focuses on the existence of ordered substructures in host structures. In addition to Ramsey’s Theorem, which gave the name to the area, another classical example of a statement of this kind is Van der Waerden’s Theorem. This theorem asserts that every coloring of positive integers with finitely many colors contains an arithmetic progression of any given length.

This paper contributes to the line of research on the existence of monochromatic configurations in coloring positive integers with finitely many colors. One of the oldest results of this kind is Schur’s Theorem from 1916: every coloring of positive integers with finitely many colors contains a monochromatic triple a, b and a+b. The theorem also implies that every such coloring contains a monochromatic triple a, b and ab. Hindman’s Conjecture offers a common generalization of these two statements: every coloring of positive integers with finitely many colors contains a monochromatic quadruple a, b, a+b and ab. We remark that Moreria [Ann. Math. 185 (2017), 1069–1090] has recently shown that any such coloring contains a monochromatic triple a, a+b and ab. The main result of this paper is that any such coloring contains a monochromatic quadruple a, b, ab and a(b+1); this disproves a recent conjecture of Sahasrabudhe.