Off-diagonal Rado number for x+y+c=z and x+qy=z

Ramsey-type problems for linear equations began with Schur’s theorem and were systematically generalized by Richard Rado. In the off-diagonal framework for two colors, one considers two different linear equations ( E 1 , E 2 ) and determines the minimum integer N for which any red–blue coloring of { 1,2 , … , N } forces either a red solution of the equation E 1 or a blue solution of the equation E 2 . In this work, we study the off-diagonal Rado numbers for a combination of homogeneous and non-homogeneous linear equations of the forms x + y + c = z and x + q y = z . We determine the exact two-color off-diagonal Rado number, denoted as R 2 ( c , q ) , associated with this system of equations.