Abstract
For n≥k≥4, let ARX+Y=Z+Tk(n) be the maximum number of rainbow solutions to the Sidon equation X+Y=Z+T over all k-colorings c:[n]→[k]. It can be shown that the total number of solutions in [n] to the Sidon equation is n3∕12+O(n2) and so, trivially, ARX+Y=Z+Tk(n)≤n3∕12+O(n2). We improve this upper bound to ARX+Y=Z+Tk(n)≤112−124kn3+Ok(n2)for all n≥k≥4. Furthermore, we give an explicit k-coloring of [n] with more rainbow solutions to the Sidon equation than a random k-coloring, and gives a lower bound of 112−13kn3−Ok(n2)≤ARX+Y=Z+Tk(n).When k=4, we use a different approach based on additive energy to obtain an upper bound of 3n3∕96+O(n2), whereas our lower bound is 2n3∕96−O(n2) in this case.