Problem 2

To make the problem a little easier, we restrict ourselves to the case ${X = {\mathbb N}}$.

An element of ${[{\mathbb N}]^\omega}$ is of the form ${x = \{x_1 < x_2 < x_3, \dots\}}$ with each ${x_i}$ in ${{\mathbb N}}$. For each ${x,y \in [{\mathbb N}]^\omega}$, define ${x \sim y}$ iff ${x_i = y_i}$ for all but finitely many ${i}$. This defines an equivalence relation on ${[{\mathbb N}]^\omega}$. Pick an element from each equivalence class. Define a ${2}$-coloring of ${[{\mathbb N}]^\omega}$ by letting ${c(x) = 0}$ if ${x}$ differs from the representative of its class on an even number of positions, and ${c(x) = 1}$ if ${x}$ differs from the representative of its class on an odd number of positions.

Can there be an infinite homogeneous subset for ${c}$?

Problem 5

Let ${(A,<)}$ be an infinite countable dense linear order without endpoints, and let ${a_1, a_2, a_3, \dots}$ be an enumeration of the elements of ${A}$. (Since ${A}$ is not a well-order, this enumeration cannot coincide with the order of ${A}$.) Similarly, let ${q_1, q_2, q_3, \dots}$ be an enumeration of ${{\mathbb Q}}$.

Construct an order-isomorphism ${f: A \rightarrow {\mathbb Q}}$ in stages. At even stages ${2s}$, make sure that ${a_s}$ is assigned a value by ${f}$. At odd stages ${2s+1}$, makes sure that ${q_s}$ is in the range of ${f}$. This technique is called back-and-forth.