Rainbow solutions to the Sidon equation

We prove that for every 4-coloring of {1,2,…,n}{1,2,…,n}, with each color class having cardinality more than (n+1)/6(n+1)/6, there exists a solution of the equation x+y=z+wx+y=z+w with x, y, z and w belonging to different color classes. The lower bound on a color class cardinality is tight.