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August 16

Induced coloring method

Where can I find information about the induced coloring method in Ramsey theory. It is used to prove the generalized Ramsey Theorem, Van der Waerdens theorem, Hales Jewett theorem etc). I want to understand the underlying idea of the method. Thanks--Shahab (talk) 06:08, 16 August 2008 (UTC)[reply]

Polynomials

How do you find the coefficients of a polynomial (of degree n-1) that passes through n points ((x₁, y₁), (x₂, y₂) . . .)? Thanks *Max* (talk) 06:22, 16 August 2008 (UTC).[reply]

Check out Polynomial interpolation. Best, RayAYang (talk) 06:31, 16 August 2008 (UTC)[reply]

Let the polynomial be

a_{0}+a_{1}x+a_{2}x^{2}\cdots a_{n-1}x^{n-1}

. Now substitute

(x_{1},y_{1})

in it. You will get the linear equation

y_{1}=a_{0}+a_{1}x_{1}+a_{2}x_{1}^{2}\cdots a_{n-1}x_{1}^{n-1}

which has n variables

a_{0},a_{1}\cdots a_{n-1}

. In this way get n linear equations and solve for the coefficients. That's actually what happens in interpolation--Shahab (talk) 06:38, 16 August 2008 (UTC)[reply]

And have also a look to the Lagrange polynomial method, that gives you immediately the interpolating polynomial, although not explicitely with its coefficients. If you need the coefficients, you can expand it and write them as linear combinations of the symmetric functions of x₁,.., x_n. But I guess this is not better than the previous method.79.38.22.37 (talk) 07:54, 16 August 2008 (UTC)[reply]

Thank you. *Max* (talk) 15:25, 16 August 2008 (UTC).[reply]