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Erdős–Turán conjecture

I don't see the revelance of the two references. No 1 (the 1936 Erdős-Turán paper) does not, repeat does not mention this statement. I can only get the first page of the No. 2 reference which specifically mentions only Erdős' name ("these results have led Erdős to conjectue...") in connection with this conjecture. So, why don't we agree that it is a conjecture of Erdős? Kope 14:25, 16 July 2007 (UTC)

The first paper is invariably cited when this conjecture is mentioned—I was trying to figure out why, since the conjecture does not (as you say) appear overtly. Certainly it is similar, and it conjectures the Green-Tao theorem (p. 264) which is related. The growth of r(n) is the real topic of the paper, but surely you agree that tight enough bounds on r(n) give the special case of the Erdős–Turán conjecture for progressions of length 3?

The second paper is explicitly about the conjecture of Erdős and Turán (conjecture first mentioned on p. 211). For what it's worth, it actually cites it to an "unpublished lecture" of Erdős rather than a paper.

If you like, I have another citation that I decided not to add, of a paper of Liangpan Li which gives a conditional proof of the conjecture. It seemed a weak result to me so I decided not to add it, but as the condition is ion a sense a different (2-dimensional) version of this conjecture perhaps it would give room for expansion?

CRGreathouse (t | c) 14:36, 16 July 2007 (UTC)

You claim: "The first paper is invariably cited when this conjecture is mentioned". Well, only recently. I do not see any reference of the divergent sum hypothesis which is earlier than, say the early seventies. Then I see zillion references by Erdos and others, all attributing it to Erdős alone. Let me add two more: R. L. Graham: Rudiments of Ramsey theory, 1980, p. 24 ("a striking conjecture of Erdős") and R. K. Guy: Unsolved problems in number theory, Springer, 1981, p. 10 ("more generally, Erdős conjectures"). Let me emphasize, that Guy and Graham are two people, who worked a lot with Erdos, were in everyday connection with him for several decades, wrote wvereal papers with him, many on this very topic, so we may assume that they are well informed. Kope 15:00, 16 July 2007 (UTC)

As I just posted on the article's Talk, Green and Tao themselves credit it to both—and they're certainly experts in the field. When I get home today I'll look in some of my books (Erdos' biography, Unsolved Problems in Number Theory, etc.) to see if I can find a mention of the conjecture.

It's not surprising that Erdos would mention that he conjectured the result, even if Turan had conjectured it with him. I've read quotes from him where he does the same without mention of another even though they make the conjecture together in a paper -- it's easy to say "I've conjectured that..." and it need not exclude the possibility of others doing the same.

Still, if you really think that Turan wasn't involved you could email some of the people who would know and see what they say to a direct question. That, or post on the NUMTHRY list.

CRGreathouse (t | c) 15:09, 16 July 2007 (UTC)

OK, the biography failed me but UPNT has something under E10:

A closely related function, with ${\displaystyle l+1=k}$, is the now famous ${\displaystyle r_{k}(n)}$, introduced long ago by Erdős & Turán: the least ${\displaystyle r}$ such that the sequence ${\displaystyle 1\leq a_{1}<a_{2}<\cdots <a_{r}\leq n}$ of ${\displaystyle r}$ numbers not exceeding ${\displaystyle n}$ must contain a ${\displaystyle k}$-term A.P.

So Guy also attributes the conjecture to the two—and Guy's been in the field for nearly as long as Erdős was. There's no reference given for this (and the only paper written by both is the one I cited), but this settles it as far as I'm concerned. Unless I get a statement to the contrary (a paper explicitly calls it a conjecture of only Erdős) I think it has to stay where it is.

CRGreathouse (t | c) 02:45, 17 July 2007 (UTC)

There are two statements.

(A) ${\displaystyle r_{k}(n)=o(n)}$.

(B) If ${\displaystyle a_{1}<a_{2}<\cdots }$ are natural numbers, ${\displaystyle \sum {\frac {1}{a_{i}}}=\infty }$, then there are arbitrarily long APs in the sequence.

I never, ever, ever, ever stated that (A) is not from Erdos and Turan. But the whole argument is not about (A), it is about (B). What I claim is that Turan had no part in conjecturing (B). It is (B), not (A) which is on the page Erdős–Turán conjecture. The whole argument is about (B), not (A). Please read the article. Kope 04:50, 17 July 2007 (UTC)

I never wrote ${\displaystyle r_{k}(n)=o(n)}$, that's all you. Guy discusses (B) by use of the function ${\displaystyle r_{k}}$, but I'm not sure if he even makes statement (A) at all.

I get the insult, though. I'm done discussing this. I suggest you get a copy of some of the standard books if you want to look further into the matter.

CRGreathouse (t | c) 12:28, 17 July 2007 (UTC)

He does. Exactly on the page you quote: "A big breakthrough was Szemerédi's proof that ${\displaystyle r_{k}(n)=o(n)}$ for all k,..." Your statement "Guy discusses (B) by use of the function ${\displaystyle r_{k}}$" is nonsense. From the fact that the density is 0, you cannot determine if the sum of the reciprocals is infinite or not. My (A) above is the Erdos-Turan conjecture, proved by Szemeredi, (B) is a conjecture of Erdos alone. Cannot be more clear. What standard books are you referring at? Kope 13:05, 17 July 2007 (UTC)

Why do you bring Szemerédi's theorem into this? Guy may mention it (I can't remember if he did in my book, version 3) but that's clearly a weaker result proved quite a time ago. CRGreathouse (t | c) 13:08, 17 July 2007 (UTC)

There is one reason I bring it into this: it is the above statement (A), conjectured by Erdos and Turan. You quote it from Guy's book and you say that it is a conjecture of Erdos and Turan. Indeed it is, however I have always spoken on a different statement, which says something else, is unsolved, and I claim that it is a conjecture of Erdos alone. You somehow confuse the two. They are very different statements. Kope 13:19, 17 July 2007 (UTC)

You, not I, wrote ${\displaystyle r_{k}(n)=o(n)}$. I didn't even use little-O notation until you did. CRGreathouse (t | c) 12:28, 18 July 2007 (UTC)

You claim that Guy attributed the statement to Erdős and Turán. For proving that you cite a sentence merely saying that they INTRODUCED a bit of notation. Whay would this support your claim? In continuing sentences Guy mentions Szemeredi's theorem. There are 2 (equivalent) ways of describing the statement of Szemeredi's theorem, one is ${\displaystyle r_{k}(n)=o(n)}$. Indeed, you never stated this, technically, you quote a sentence simply saying that E. and T. defined a function. Defining a function is not a conjecture. There are two statements. They are different. One is a theorem, the other is a conjecture. We should not confuse them. Kope 12:48, 19 July 2007 (UTC)

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I've started an article, ideal (set theory), which is where I think small set should probably eventually be merged. Right now I have no lack of material to add to it, but I haven't figured out just how it should be organized. --Trovatore 22:33, 5 August 2007 (UTC)

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Thanks! Dreadstar † 01:45, 11 October 2007 (UTC)

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User:Mikkalai/By Erdos contains a very raw list made from remnants of categories and the log of the bot which implemented the deletion you opposed. Please join the discusion here to decide how to proceded. A clandestinely proud Erdos-Number-3-wikipedian `'Míkka 16:28, 2 November 2007 (UTC)

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You're a longstanding Misplaced Pages contributor with a solid history. User:VoteFair is a single-purpose account with an agenda of self-promotion. And yet VoteFair claims that you approve of his edits and that you will re-add content for him, and you haven't been refuting this idea. What's the deal? rspeer / ɹəədsɹ 08:34, 13 November 2007 (UTC)