The Anti-Ramsey Problem for the Sidon equation

Abstract

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 $n^3/12+O(n^2)$ and so, trivially, $A{R}_{X+Y=Z+T}^k(n)\le n^3/12+O(n^2)$. We improve this upper bound to \[ AR_{X+Y = Z+ T}^k (n) \leq \left( \frac{1}{12} - \frac{1}{24k} \right)n^3 + O_k(n^2) \] 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 \[ \left( \frac{1}{12} - \frac{1}{3k} \right)n^3 - O_k (n^2) \leq AR_{X+Y = Z+ T}^k (n). \] When $k=4$, we use a different approach based on additive energy to obtain an upper bound of $3n^3/96+O(n^2)$, whereas our lower bound is $2n^3/96-O(n^2)$ in this case.