## Sum-free sets:

A subset $$A'$$ of an abelian group $$(A,+)$$ is sum-free if there are no elements $$a,b,c \in A'$$ with $$a+b = c$$.

## Theorem:

Every set of $$n$$ non-zero integers contains a sum-free set of size at least $$\frac{n}{3}$$.

## Proof:

Let $$A$$ be a set of non-zero integers with $$\lvert A \rvert = n$$. Now, if we choose a real number $$\theta \in [0,1]$$ we may define:

\begin{equation} A_{\theta} = \{a \in A | a \cdot \theta - \lfloor a \cdot \theta \rfloor \in (\frac{1}{3}, \frac{2}{3}) \} \end{equation}

and we’ll note that $$A_{\theta}$$ is always sum-free since:

\begin{equation} \forall a, b, c \in (\frac{1}{3}, \frac{2}{3}), a + b \neq c \end{equation}

Moreover, if we sample uniformly from $$[0,1]$$ we will observe that:

\begin{equation} P(a \in A_{\theta}) = \frac{1}{3} \end{equation}

wince $$a_{\theta} \sim U([0,a])$$. This implies that for any $$A \subset \mathbb{N}^*$$ there is a sum-free subset with size at least $$\frac{\lvert A \rvert}{3}$$.

1. Noga Alon & Joel Spencer. The probabilistic method. John Wiley & Sons. 2000.

2. Yufei Zhao. Probabilistic Methods in Combinatorics. MIT. 2020.