Talk:Borsuk's conjecture

What is c?

Quoting from the article (version 339480200, January 2010):

The proof by Kahn and Kalai implies that for large enough d, one needs $\alpha (d)>c^{\sqrt {d}}$ number of pieces.

The constant $c$ isn't mentioned prior to that and the mention after that sentence doesn't make it any clearer either. 92.225.32.151 (talk) 04:05, 25 March 2010 (UTC)

Well, it is explained in last words of the paragraph: ...for some c > 1.
It is simply some constant. It is not important here what constant it is. Borsuk asked whether the number of pieces needed equals the space dimension plus one, and K&K have proven the $\alpha (d)$ function grows faster than some function growing faster than any linear function. This implies $\alpha (d)$ grows faster than a linear function, consequently for some $d$ big enough $\alpha (d)>d+1$. That's enough, that is an answer (negative) for the Borsuk's problem. They didn't need a precise bounding for $\alpha (d)$, so the precise value of $c$ does not matter, as long as it is greater than 1. --CiaPan (talk) 08:43, 25 March 2010 (UTC)

No. That doesn't have anything to do with it. The sentence that you're referring to is

It is conjectured (see e.g. Alon's article) that $\alpha (d)>c^{d}$ for some c > 1.

That is a conjectured strengthening of what's said in the sentence before that (and has been proven):

The proof by Kahn and Kalai implies that for large enough d, one needs $\alpha (d)>c^{\sqrt {d}}$ number of pieces.

Because if $c$ is greater than 1, $c^{d}$ will be greater than $c^{\sqrt {d}}$. However, the constant $c$ in those two statements have nothing to with one another! The fact that they bear the same name is mere coincidence!

If the value of that constant doesn't matter, the article should say so! 92.225.37.13 (talk) 17:52, 25 March 2010 (UTC)

Imagine I know what I said—the clause '...for some c > 1' applies to BOTH inequalities.

That's not clear from the wording at all! 85.179.71.78 (talk) 07:32, 26 March 2010 (UTC)

Kahn and Kalai have shown their inequality for a specific value of c=1.203, BUT that value in fact does NOT matter, at least for the so called Borsuk's conjecture. For ANY $c>1$, if $\alpha (d)>c^{\sqrt {d}}$, then there exist such (big enough) d, that $\alpha (d)>d+1$. Whatever small c is, as long as it is greater than 1, the inequality shown disproves the conjecture. --CiaPan (talk) 21:41, 25 March 2010 (UTC)

My problem is not with understanding the implication of what was shown and how the exact value of $c$ doesn't matter. My point is that the parser in my head, when reading that article, says "i don't know what $c$ is"! —Preceding unsigned comment added by 85.179.71.78 (talk) 07:34, 26 March 2010 (UTC)

Please see this edit and comment if it made the article clear enough.
{{Copyedit}}–ing would be appreciated. --CiaPan (talk) 20:27, 7 April 2010 (UTC)

'order of magnitude'

• http://wiki.riteme.site/w/index.php?title=Borsuk%27s_conjecture&diff=prev&oldid=361693685

Why is 'rate of growth' better than 'order of magnitude'? Noga Alon uses the latter in his work (arXiv:math/0212390v1, PDF, PS), do we (esp. 66.36.154.45) know the math language better than him? --CiaPan (talk) 15:54, 12 May 2010 (UTC)

Help me with wording

In a note I added in the lead section (see diff):

Karol Borsuk has formulated the problem just as a question

wouldn't it be better to use 'expressed', or maybe 'worded' or 'posed' instead of 'formulated'...?

Non native speaker, CiaPan (talk) 09:02, 7 June 2017 (UTC)
(please reply here – I'm watching this article)

I think 'posed' would work betting in place of 'formulated', though formulated makes sense as well. RickinBaltimore (talk) 13:48, 7 June 2017 (UTC)

OK, so I'll leave it just 'as is'. Possibly somebody will improve it some day, but I just needed to know if it needs a fix right now.
Thank you, RickinBaltimore, for sharing your opinion. --CiaPan (talk) 14:16, 7 June 2017 (UTC)

article title

Should the article be retitled if the name "Borsuk's conjecture" is incorrect? Eric Rowland (talk) 16:48, 30 March 2019 (UTC)

Hi, Eric Rowland! The name is essentially incorrect, because the statement was considered 'a conjecture' by many, but not by Karol Borsuk. However, this name is commonly used and encyclopedia has to report a common knowledge, not fix it. As a Pole I would love to rename the article, too, to represent the historical truth (similar to the corresponding article in Polish Wikipedia, which is a Geometric problem by Karol Borsuk, pl:Problem geometryczny Karola Borsuka). However, that wouldn't be recognized in English-speaking world, which uses 'conjecture', so I just added some notes on 'correctnes' and 'historical reasons', and that's all. If you wish to see it changed, you'd have to begin a battle for a name change in reputable mathematical journals, an when you win it (many mathematicians start to use a new name) then Wikipedia will be able to change its naming, too. But I don't expect it to happen sooner than in a couple of decades... :) --CiaPan (talk) 16:30, 31 March 2019 (UTC)

Overwhelmingly unclear sentence

After mentioning counterexamples found by Kalai & Kahn, the article says:

"However, as pointed out by Bernulf Weißbach, the first part of this claim is in fact false..

It is overwhelmingly unclear what "the first part of this claim" means. It would be so much better if the article simply said just exactly what it is that is false.98.255.224.144 (talk) 23:17, 17 January 2021 (UTC)

This claim is quite obviously what the previous sentence reports; namely, what Kahn & Kalai claimed. And their claim was “that their construction shows that n + 1 pieces do not suffice for n = 1325 and for each n > 2014.” There are two disjoint parts of it:

1. “that their construction shows that n + 1 pieces do not suffice for n = 1325,” and
2. “that their construction shows that n + 1 pieces do not suffice for each n > 2014.”

The former turned out to be false: their construction did not show that n + 1 pieces do not suffice for n = 1325.

See the linked paper by Bernulf Weißbach, page 418. --CiaPan (talk) 02:38, 18 January 2021 (UTC)