The rainbow saturation number is linear

Given a graph H, we say that an edge-colored graph G is H-rainbow saturated if it does not contain a rainbow copy of H, but the addition of any nonedge in any color creates a rainbow copy of H. The rainbow saturation number rsat(n, H) is the minimum number of edges among all H-rainbow saturated edge-colored graphs on n vertices. We prove that for any nonempty graph H, the rainbow saturation number is linear in n, thus proving a conjecture of Girao, Lewis, and Popielarz. In addition, we give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Girao, Lewis, and Popielarz.