On the rainbow Cameron-Erd\H{o}s problem with respect to generalized Sidon sets of multidimensional grids

TL;DR

This paper investigates the maximum number of r-colorings of multidimensional grids avoiding rainbow solutions to a generalized linear equation, extending known results and confirming conjectures in combinatorics.

Contribution

It provides asymptotic counts for such colorings, characterizes typical colorings, and proves uniqueness of the grid in maximizing these colorings, connecting to several open problems.

Findings

Asymptotic number of r-colorings without rainbow solutions is obtained.

Typical colorings avoiding rainbow solutions are (kh-1)-colorings.

The grid [n]^d uniquely maximizes the number of such colorings among all subsets.

Abstract

For positive integers $n$, $d$, $k$ and $h$, let $[n]^d$ be the $d$-dimensional grid of order $n$, and we refer to the equation $\sum_{i=1}^{h}x_{1,i}=\cdots =\sum_{i=1}^{h}x_{k,i}$ as the {\it $B_{k,h}$-equation}, where $x_{1,1}, \ldots, x_{1,h}, \ldots, x_{k,1}, \ldots, x_{k,h}$ are $kh$ points in $[n]^d$. In this paper, we study the rainbow Cameron-Erd\H{o}s problem with respect to the $B_{k,h}$-equation. We obtain the asymptotic number of $r$-colorings of $[n]^d$ without rainbow solutions to the $B_{k,h}$-equation, and we show that the typical colorings with this property are $(kh-1)$-colorings. We also prove that among all subsets of $[n]^d$, $[n]^d$ is the unique subset admitting the maximum number of $r$-colorings without rainbow solutions to the $B_{k,h}$-equation. The case $d=1$ and $k=h=2$ of our result confirms a conjecture on Sidon sets by Lin, Wang and Zhou~[{\it European J. Combin.}, 2022]; the case $d=1$, $k=2$ and $h\geq 2$ of our result partly solves a problem concerning linear equations proposed by Cheng, Jing, Li, Wang and Zhou~[{\it J. Combin. Theory Ser. A}, 2023]; the case $d\geq 2$ and $k=h=2$ corresponds to colorings without rainbow (possibly degenerate) parallelograms, and this geometric perspective might be of independent interest. Our proof combines the hypergraph container method with a stability analysis and a deviation gain argument.