Problem

Source: Romania JBMO TST 2022

Tags: combinatorics, Romanian TST



We call a set $A\subset \mathbb{R}$ free of arithmetic progressions if for all distinct $a,b,c\in A$ we have $a+b\neq 2c.$ Prove that the set $\{0,1,2,\ldots 3^8-1\}$ has a subset $A$ which is free of arithmetic progressions and has at least $256$ elements.