The Anti-Ramsey Problem for the Sidon equation

For $n\ge k\ge 4$, let $A{R}_{X+Y=Z+T}^k(n)$ be the maximum number of rainbow solutions to the Sidon equation $X+Y=Z+T$ over all $k$-colorings $c:[n]\to [k]$. It can be shown that the total number of solutions in $[n]$ to the Sidon equation is $n^3\mathrm{/}12+O(n^2)$ and so, trivially, $AR_{X+Y = Z + T}^k (n) \leq n^3 /12 + O (n^2)$. We improve this upper bound to $$A{R}_{X+Y=Z+T}^k(n)\le (\frac{1}{12}-\frac{1}{24k})n^3+O_k(n^2)$$ for all $n\ge k\ge 4$. 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 $$(\frac{1}{12}-\frac{1}{3k})n^3-O_k(n^2)\le A{R}_{X+Y=Z+T}^k(n)\mathrm{.}$$ When $k=4$, we use a different approach based on additive energy to obtain an upper bound of $3n^3 / 96 + O(n^2)$,whereasourlowerboundis$2n^3 / 96 - O (n^2)$ in this case.