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If $n$ is a natural number let $f(n)$ denote the largest number of squares in an AP of length $n$ (consisting of integers). As there is no AP of length 4 consisting of squares (Euler), $f(n)\leq r_4(n)$. By Szemeredi's theorem for 4-AP, $r_4(n)=o(n)$, so $f(n)=o(n)$. In fact, this was Szemeredi's orginal motivation for his theorem.