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I have been studying erdős conjecture on arithmetic progressions for some time and have an interesting question for you :

How do I strictly define "a set containing arithmetic progressions of any given length"

And can somebody give me an example of such a set?

Thank you in advance, you guys are awesome

Thomas
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  • set containing all possible arithmetic progressions of any length or just set which contains some arithmetic progression for any given length? – Mihail Aug 17 '15 at 15:05
  • For any given $k \in \mathbb N$, the set contains an arithmetic progression of length $k$. An example of such a set is $\mathbb N$ itself, which contains the arithmetic progression $1,2,\ldots,k$. There are more interesting examples, such as the set of all primes. I don't see what you have been studying about the Erdős conjecture without actually understanding the statement of the conjecture itself? – Erick Wong Aug 17 '15 at 15:05
  • Hi Erick and Mihail, thank you for the quick response! Erick one more question, is it "an arithmetic progression" or "infinite many arithmetic progressions"? – Thomas Aug 17 '15 at 16:11
  • @Thomas It makes no difference. If a set contains an arithmetic progression of each finite length, then it contains infinitely many arithmetic progressions of each finite length. Just think about how an arithmetic progression of length $100$ contains several arithmetic progressions of length $10$. What have you studied about this conjecture so far? – Erick Wong Aug 18 '15 at 07:37

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