Question 108
We say that a set $A \subset \mathbb{R}$ with at least three elements is free of arithmetic progressions if for any distinct $a, b, c \in A$ we have $a+b \neq 2c$.
Show that the set $\{0,1,2, \ldots, 3^{8}-1\}$ contains a subset $A$ with at least 256 elements that is free of arithmetic progressions.
Show that the set $\{0,1,2, \ldots, 3^{8}-1\}$ contains a subset $A$ with at least 256 elements that is free of arithmetic progressions.