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Hints Homework 4

September 14, 2011

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.

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