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arXiv · 1903.04132

Maximum $k$-sum $\mathbf{n}$-free sets of the 2-dimensional integer lattice

Ilkyoo Choi·Ringi Kim·Boram Park
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Abstract

For a positive integer $n$, let $[n]$ denote $\{1, \ldots, n\}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, a \textit{$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$ is a subset $S$ of $[n]\times [n]$ such that there are no elements ${\mathbf{a}}_1, \ldots, {\mathbf{a}}_k$ in $S$ satisfying ${\mathbf{a}}_1+\cdots+{\mathbf{a}}_k =\mathbf{b}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, we determine the maximum density of a {$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$. This is the first investigation of the non-homogeneous sum-free set problem in higher dimensions.

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Ilkyoo Choi, Ringi Kim, Boram Park. 2019-03-12. Maximum $k$-sum $\mathbf{n}$-free sets of the 2-dimensional integer lattice. https://arxiv.org/abs/1903.04132

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