---
tag: 7SHV2SPYP6
kind: concept
name: Sum-free set
msc: [11B75, 11P70, 05D10]
sources:
  - url: https://en.wikipedia.org/wiki/Sum-free_set
same_as:
  wikidata: Q3054917
  wikipedia: Sum-free set
---

## Summary

A sum-free set is a subset $A$ of an abelian group that contains no solution of $x + y = z$ with $x, y, z \in A$.
Equivalently, the [[sumset|XNBN31HQWB]] $A + A$ is disjoint from $A$.
Sum-free sets are central in [[additive combinatorics|3NM362A2JG]] and [[Ramsey theory|G5TWXV565S]].
[[Schur's theorem|V6XJQXZ7E8]] says that the positive integers cannot be split into finitely many sum-free sets.

## Explanations

### L1 human

A set of numbers is sum-free if adding two of its members, possibly the same one twice, never gives a member.
The odd numbers are sum-free, because two odd numbers add up to an even number.
The set $\set{6, 7, 8, 9, 10}$ is sum-free too, because every sum of two of its members is at least $12$.
The set $\set{1, 2, 3}$ is not, since $1 + 2 = 3$.