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.