# Hints Homework 4

**Problem 2**

To make the problem a little easier, we restrict ourselves to the case .

An element of is of the form with each in . For each , define iff for all but finitely many . This defines an equivalence relation on . Pick an element from each equivalence class. Define a -coloring of by letting if differs from the representative of its class on an even number of positions, and if differs from the representative of its class on an odd number of positions.

Can there be an infinite homogeneous subset for ?

**Problem 5**

Let be an infinite countable dense linear order without endpoints, and let be an enumeration of the elements of . (Since is not a well-order, this enumeration cannot coincide with the order of .) Similarly, let be an enumeration of .

Construct an order-isomorphism in stages. At even stages , make sure that is assigned a value by . At odd stages , makes sure that is in the range of . This technique is called *back-and-forth*.