Talk:Szemerédi's theorem

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In the current version of the entry it is written that: 'positive logarithmic density', i.e. a sequence whose series of reciprocals diverges. In my opinion divergent series of reciprocals is not the same as positive logarithmic density. (For the definition of the logarithmic density, which I know, see [1]) For instance, the set ${\displaystyle P}$ of all primes has asymptotic density (hence also logarithmic density) equal to 0, but the series of reciprocals is divergent. The conjecture was stated with reciprocals. Have you seen another definition of logarithmic density elsewhere? --Kompik 18:18, 6 October 2005 (UTC)[reply]

I have removed the phrase. Charles Matthews 20:13, 6 October 2005 (UTC)[reply]

The 1936 Erdős-Turán paper does not mention the conjecture that if the sum of reciprocals diverges then the sequence contains arbitrarily long APs. I don't know when the problem was first raised. Incidentally, Erdős mentions in a 1961 paper (P.Erdős: Some unsolved problems, Magyar Tud. Akad. Mat. Kut. Int. Közl., 9(1961), 221-254.) that the Erdős-Turán problem (i.e., the Szemerédi theorem) was raised by Issai Schur around 1930. Kope 15:42, 12 July 2007 (UTC)[reply]

I'm willing to believe you're correct. Could you fix this article, also? — Arthur Rubin | (talk) 17:39, 18 July 2007 (UTC)[reply]

In the same 1936 paper, Erdős and Turán conjectured^[1] that 'positive density' could here be relaxed to any sequence whose series of reciprocals diverges (see small set). This would in particular apply to the series of prime numbers, implying the corollary

the sequence of primes numbers contains arithmetic progressions of any length.

This result (only the corollary, not the Erdős-Turán conjecture) was proven by Ben Green and Terence Tao in 2004 and is now known as the Green-Tao theorem.

So, the above piece removed until clarification. `'Míkka 02:26, 19 July 2007 (UTC)[reply]

There is something strange about this quotation. Not that Graham did not say, but Erdős used it before even Szemeredi proved the general result...: "Rencetly, Szemeredi proved ${\displaystyle r_{4}(n)=o(n)}$; his very complicated proof is a masterpiece of combinatorial reasoning." This is just a special case, with presumably much simpler proof than the great theorem!

P. Erdős:Problems and results in combinatorial number theory, A Survey of Comb. Th., Proc. Int. Symp. Colorado State Univ. Fort Collins, CO, pp. 117-138, North-Holland 1973. Kope 17:27, 19 July 2007 (UTC)[reply]

I've changed the attribution to Erdős and added a reference. Searching for the phrase turns up many publications. I'm not sure it's worth tracking down who exactly published this phrase first. Gutworth (talk) 03:05, 4 June 2015 (UTC)[reply]

C > 0 - doesn't it mean C > 1? —Preceding unsigned comment added by 131.111.213.44 (talk) 16:36, 2 June 2008 (UTC)[reply]