For every integer \( c \), let \( r(2, 2, c) \) be the least integer \( n \) such that for every 2-coloring of the set \( \{1, 2, \ldots, n\} \) there exists a monochromatic solution to the equation \[ 2x_1 + 2x_2 + c = x_3. \]
Secondly, for every integer \( c \), let \( r(2, 2, 2, c) \) be the least integer \( n \) such that for every 2-coloring of the set \( \{1, 2, \ldots, n\} \) there exists a monochromatic solution to the equation \[ 2x_1 + 2x_2 + 2x_3 + c = x_4. \]
In this paper, exact values are found for \( r(2, 2, c) \) and \( r(2, 2, 2, c) \).