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# ps5 - Probability and Algorithms Leonard J Schulman Problem...

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Probability and Algorithms Caltech CS150, Winter 2006 Leonard J. Schulman TA: Chih-Kai (Kevin) Ko. Problem set 5 Out Wednesday February 22. Due Monday March 6. Week 6 reading: MU 6.7-6.8, 7.1-7.4, proof of the Perron-Frobenius theorem in notes of Andries Brouwer on the class website. This assignment is longer than usual so I’m allowing two weeks for it. Please get started early though. I haven’t assigned work on the Perron-Frobenius theorem but it is an important theorem with an interesting proof, and well worth learning. (You can also find the proof elsewhere, e.g., some notes online by Mike Boyle.) Exercises: MU 6.16, 6.18, 7.1, 7.6, 7.8, 7.13, 7.17, 7.19, 7.26. Also the following two exercises: (A) Let the van der Waerden function W ( k ) , for nonnegative integer k , be the least nonnegative integer n so that if { 1 , . . . , n } is two-colored, there exists a monochromatic arithmetic progression with k terms. Remarkably, W ( k ) is finite for all k ; the upper bound is immense and the true value is far from known. In this exercise you are asked to show a lower bound on W ( k ) . Try coloring
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