The Anti-Ramsey Problem for the Sidon equation (1808.09846v1)
Abstract: For $n \geq k \geq 4$, let $AR_{X + Y = Z + T}k (n)$ be the maximum number of rainbow solutions to the Sidon equation $X+Y = Z + T$ over all $k$-colorings $c:[n] \rightarrow [k]$. It can be shown that the total number of solutions in $[n]$ to the Sidon equation is $n3/12 + O(n2)$ and so, trivially, $AR_{X+Y = Z + T}k (n) \leq n3 /12 + O (n2)$. We improve this upper bound to [ AR_{X+Y = Z+ T}k (n) \leq \left( \frac{1}{12} - \frac{1}{24k} \right)n3 + O_k(n2) ] for all $n \geq k \geq 4$. 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)n3 - O_k (n2) \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 $3n3 / 96 + O(n2)$, whereas our lower bound is $2n3 / 96 - O (n2)$ in this case.