Question 191
$$
S_n = \{(x, y) \in \mathbb{Z}^2 \mid 1 \le x, y \le 5n\}.
$$
A subset $A \subseteq S_n$ is said to be *sumfree* if
$$
(x_1, y_1), (x_2, y_2) \in A \implies (x_1 + x_2, y_1 + y_2) \notin A.
$$
Show that there exists a sumfree subset with at least $15n^2$ elements.
S_n = \{(x, y) \in \mathbb{Z}^2 \mid 1 \le x, y \le 5n\}.
$$
A subset $A \subseteq S_n$ is said to be *sumfree* if
$$
(x_1, y_1), (x_2, y_2) \in A \implies (x_1 + x_2, y_1 + y_2) \notin A.
$$
Show that there exists a sumfree subset with at least $15n^2$ elements.