The Anti-Ramsey Problem for the Sidon equation

TL;DR

This paper investigates the maximum number of rainbow solutions to the Sidon equation in k-colorings of [n], providing improved upper bounds, explicit colorings with many solutions, and bounds for specific cases like k=4.

Contribution

It offers new upper bounds on rainbow solutions, constructs explicit colorings surpassing random colorings, and applies additive energy methods for the case k=4.

Findings

Improved upper bound: rac{1}{12} - rac{1}{24k} for all n igg; k igg; 4.

Explicit colorings with more solutions than random colorings.

Specific bounds for the case k=4 using additive energy.

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 $n^3/12 + O(n^2)$ and so, trivially, $AR_{X+Y = Z + T}^k (n) \leq n^3 /12 + O (n^2)$. We improve this upper bound to \[ AR_{X+Y = Z+ T}^k (n) \leq \left( \frac{1}{12} - \frac{1}{24k} \right)n^3 + O_k(n^2) \] for all $n \geq k \geq 4$. 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 \[ \left( \frac{1}{12} - \frac{1}{3k} \right)n^3 - O_k (n^2) \leq AR_{X+Y = Z+ T}^k (n). \] 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.