Wikipedia:Reference desk/Archives/Mathematics/2008 August 31

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August 31

Problem in a proof

I was reading about Ramsey theory on this page and am stuck on a seemingly trivial point in the proof of Van der Waerden's theorem (It is Theorem 6 in the pdf, page 26 or 28). The problem is this: First we define a function $\chi :\{1,\cdots M(M'+1)\}\rightarrow \{1,\cdots r\}$. Then the writer claims that this induces a function $\chi ':\{1,\cdots M'\}\rightarrow \{1\dots r^{M}\}$ such that the following condition holds:

$\chi '(k)=\chi '(k')\iff \chi (kM+j)=\chi (k'M+j)\forall j\in \{1\cdots M\}$

What I don't understand is how is the existence of such a function guaranteed. Why is $r^{M}$ large enough to accomodate the various values that numbers from 1 to M' will take while retaining the imposed condition. (Earlier he had stated that M' was the Van der Waerden number for l and $r^{M}$ i.e. if numbers from 1 to M' are assigned into classes from 1 to $r^{M}$ at least l numbers would fall in the same class. I do not know whether this has anything to do with my question.)

I would appreciate any help. Thanks.--Shahab (talk) 07:30, 31 August 2008 (UTC)

Here's my impression of it: he wants you to take the long sequence of colors in blocks, and treat each smaller sequence of colors within a block like a single more complicated color. So, say the integers 1-20 were colored with the numbers 1-3. We could have, for instance, 13112131221213233312. You could imagine this instead as coloring the integers 1-5 with the numbers 1111-3333 where the digits are all 1's, 2's and 3's. In other words, (1311)(2131)(2212)(1323)(3312). Since there are 3^4 colors of that form, we could relabel them with the numbers 1-81. Two integers are colored the same precisely when their blocks of colors are the same, which in symbols is what he wrote. He also seems to leave off a bit at the beginning of the original sequence, since he doesn't need all of it. You'll notice his condition only applies to terms in the original sequence with index at least M+1. Black Carrot (talk) 23:45, 2 September 2008 (UTC)

Sine Law Ambiguous Case

Hello. Are imaginary solutions possible if a < b sin A like if the discriminant in the quadratic formula is negative? Thanks in advance. --Mayfare (talk) 19:20, 31 August 2008 (UTC)

I do not understand your question. a cannot be smaller than b sin A. Secondly what quadratic formula? I do not see any quadratic formula. I think you assumed that other people know what you know about the problem. If there is a method of determining a than you may have to show us. 202.147.44.80 (talk) 22:56, 31 August 2008 (UTC)

OK. I am assuming that you are looking at the equation $\scriptstyle \sin B=(b/a)\sin A$ which comes from the sine law. If $\scriptstyle a<b\sin A$ then there are no real solutions for B to this equation. Yes, there would be solutions in the complex plane, but they would not be of interest if you are looking at the geometric problem (though they would probably be relevant to some appropriate geometric generalization). And you are right, that this is analogous to the situation with the quadratic formula, in the sense that many equations that have no real solutions will have complex solutions. Oded (talk) 23:02, 31 August 2008 (UTC)

Geometrical problems often lead to equations where only the real solutions make sense. The equation sin x = a has real solutions when a is a real number and |a| ≤ 1, say, a = 1/2. Otherwise it has non-real complex solutions that cannot be interpreted as angles. The equation sin x = 2 has a solution x ≈ 1.571 + 1.317i. Bo Jacoby (talk) 11:39, 1 September 2008 (UTC).