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I'm looking for some clarity on the statement mainly, that I haven't been able to find anywhere else. I'll illustrate my question with an example:

Consider the set [itex]A=\{1,3,5,7,9,11\}[/itex]. This set contains an arithmetic progression of length 6, but can we also say it contains arithmetic progressions of length 5, 4, and 3?

In other words, is the statement "a set of natural numbers does not contain arbitrarily long arithmetic progressions" equivalent to the statement "there exists some natural number [itex]N[/itex] such that the set contains no arithmetic progressions of length greater than [itex]N[/itex]"?