Shape of the asymptotic maximum sum-free sets in integer lattice grids

TL;DR

This paper characterizes the shape of nearly maximum sum-free subsets in two-dimensional integer grids, showing they are confined within a specific stripe and determining maximum sizes for subsets avoiding certain linear triples.

Contribution

It precisely describes the asymptotic shape of maximum sum-free sets in 2D grids and extends to subsets avoiding linear triples of the form px+py=z.

Findings

Sum-free sets of size close to 3/5 n^2 lie within a specific stripe.

The shape of maximum sum-free sets is fully characterized asymptotically.

Maximum size of subsets avoiding triples with linear relation px+py=z is determined.

Abstract

We determine the shape of all sum-free sets in $\{1,2,\ldots,n\}^2$ of size close to the maximum $\frac{3}{5}n^2$, solving a problem of Elsholtz and Rackham. We show that all such asymptotic maximum sum-free sets lie completely in the stripe $\frac{4}{5}n-o(n)\le x+y\le\frac{8}{5}n+ o(n)$. We also determine for any positive integer $p$ the maximum size of a subset $A\subseteq \{1,2,\ldots,n\}^2$ which forbids the triple $(x,y,z)$ satisfying $px+py=z$.