Exercise 4, § 2

Let $n\geqslant 1$ be an integer. We note $S(n)$ the largest integer $k\geqslant 0$ such that, for every set X of $n$ real numbers, there exists a subset $Y\subset X$ of cardinal $k$ such that the equation $x+y=z$ admits no solution in Y. We therefore have $S(n)\leqslant n$.

a) Now prove that the limit

$\sigma =$ lim$_{n\rightarrow+\infty}\frac{S(n)}{n}$

exists.

b) Now prove that $\sigma \leqslant 2/5$.

c) Now prove that $\sigma \geqslant 1/3$.

(One can prove that $\sigma = 1/3$, cf. S. Eberhard, B. Green, F. Manners, Sets of integers with no large sum-free subset, Annals of Mathematics 180 (2014), 621–652.)