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On the number of sum-free subsets of the square grid

Anubhab Ghosal

TL;DR

This work resolves the two-dimensional Cameron–Erdős-type conjecture by showing the number of sum-free subsets of $[n]^2$ satisfies $\sf([n]^2)=2^{0.6n^2+O(n)}$. The authors combine the hypergraph container method with Green's Removal Lemma and stability to first show that almost all sum-free sets lie near the extremal stripe (proximity), and then perform a delicate fiber-based counting analysis of regions near the stripe (direction count). A key innovation is the decomposition into 1D fibers and a height-profile framework that controls discrepancies and inter-fiber interactions, yielding the sharp exponential growth rate. The results extend the Cameron–Erdős program to two dimensions, align with the known lower-bound stripe construction, and lay groundwork for higher-dimensional analogues via a similar container-fiber strategy.

Abstract

Generalising the Cameron--Erdős conjecture to two dimensions, Elsholtz and Rackham conjectured that the number of sum-free subsets of $[n]^2$ is $2^{0.6n^2+O(n)}$. We prove their conjecture.

On the number of sum-free subsets of the square grid

TL;DR

This work resolves the two-dimensional Cameron–Erdős-type conjecture by showing the number of sum-free subsets of satisfies . The authors combine the hypergraph container method with Green's Removal Lemma and stability to first show that almost all sum-free sets lie near the extremal stripe (proximity), and then perform a delicate fiber-based counting analysis of regions near the stripe (direction count). A key innovation is the decomposition into 1D fibers and a height-profile framework that controls discrepancies and inter-fiber interactions, yielding the sharp exponential growth rate. The results extend the Cameron–Erdős program to two dimensions, align with the known lower-bound stripe construction, and lay groundwork for higher-dimensional analogues via a similar container-fiber strategy.

Abstract

Generalising the Cameron--Erdős conjecture to two dimensions, Elsholtz and Rackham conjectured that the number of sum-free subsets of is . We prove their conjecture.
Paper Structure (15 sections, 25 theorems, 26 equations, 2 figures)

This paper contains 15 sections, 25 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Sum-free subsets of $[n]$
  3. Sum-free sets in two dimensions
  4. Higher dimensions
  5. Proof strategy and organisation
  6. Approximate typical structure
  7. Containers
  8. Green's Removal Lemma
  9. Proof of Proposition \ref{['proximity']}
  10. Counting sum-free sets close to the big stripe
  11. A different region
  12. Decomposing into fibers
  13. Bounding $\sf(\{1,3,4\}\times [n])$
  14. Proof of Lemma \ref{['bounding g']}: Bounding $g(n)$.
  15. Height Profile

Key Result

Theorem 1.3

The maximal size of a sum-free subset of $[n]^2$ is $M([n]^2)=0.6n^2+O(n)$.

Figures (2)

  • Figure 1: The set of lattice points in the big stripe is sum-free.
  • Figure 2: The simpler region $R \cup L$

Theorems & Definitions (61)

  • Definition 1.2
  • Theorem 1.3: elsholtz2017maximal
  • Definition 1.4
  • Theorem 1.5: liu2023shape
  • Conjecture 1.6: elsholtz2017maximal
  • Theorem 1.7
  • Conjecture 1.8
  • Conjecture 1.9: elsholtz2017maximal
  • Proposition 1.10
  • Proposition 2.1
  • ...and 51 more