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Hints Homework 9

November 2, 2011

Problem 1

First argue that it suffices to prove the result for {n} of the form {2^k}. Then proceed by induction: Assume {(\pi(1), \dots, \pi(n))} is a permutation of {(1, \dots, n)} that does not contain a monotone {3}-AP. Consider the sequence {(2\pi(1)-1, 2\pi(2)-1, \dots, 2\pi(n)-1, 2\pi(1), 2\pi(2), \dots, 2\pi(n))}.

Problem 2

Choose {n \geq HJ(k,r)} and a prime number {p > k}. Consider the set

\displaystyle  	A = \{a_0 + a_1p + \cdots + a_{n-1}p^{n-1}\colon 0 \leq a_i < k \}.

Problem 3

Find a way how you can identify {C^{ns}_t} with {C^s_{t^n}}, so that a line in {C^s_{t^n}} can be identified with an {n}-dimensional subspace of {C^{ns}_t}. Then apply the Hales-Jewett Theorem.

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