If Erdős Conjecture on arithmetic progressions is true, and $A$ is large, then does there exist a consecutive A.P. of $A$ of length $k$ for every $k?$

Question

Question 1: If Erdős Conjecture on arithmetic progressions is true, and $A\subset \mathbb{N}$ is large, then does $A$ contain a consecutive A.P. of $A$ of length $k;\ \lbrace{a_{m+1}, a_{m+2},\ldots, a_{m+k}\rbrace},\ $ for every $k\in\mathbb{N}?$

Maybe this is true by Roth's theorem?