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Has Erdős' conjecture on arithmetic progressions or his conjecture on Sylvester's sequence been proven?

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    $\begingroup$ @BrianM.Scott I think "Erdős's conjectures" would be more correct, though. The fact that s is pronounced as sh here suggests that the possessive form should be treated in the same way as "Bush's". $\endgroup$ – user53153 Jan 22 '13 at 2:45
  • $\begingroup$ To the best of my knowledge, the 1st conjecture remains open. I don't know about the second one. $\endgroup$ – Gerry Myerson Jan 22 '13 at 3:09
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    $\begingroup$ Reminds me of a limerick I saw over forty years ago $\endgroup$ – marty cohen Jan 22 '13 at 4:54
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The first conjecture is wide open. It is - roughly speaking - equivalent to proving Szemeredi's theorem with extremely good bounds. Even if you just ask for progressions of length $3$, then the best known bounds are not strong enough. If you start talking about arbitrary length, the bounds become appropriately worse.

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