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+1 for spelling his name right! –  Brian M. Scott Jan 22 '13 at 2:41
@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". –  user53153 Jan 22 '13 at 2:45
To the best of my knowledge, the 1st conjecture remains open. I don't know about the second one. –  Gerry Myerson Jan 22 '13 at 3:09
Reminds me of a limerick I saw over forty years ago –  marty cohen Jan 22 '13 at 4:54

1 Answer 1

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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