# Hints HW 6

**Problem 1**

This is the same argument as in class. Just replace by .

**Problem 2**

Again, generalize the argument from class. Consider the lexicographic ordering on , i.e. iff where is the least ordinal such that . This is a linear order on .

Argue that with this ordering has no infinite decreasing or increasing sequence of length . For this, you have to “lift” the Pigeonhole Principle analysis to binary sequences of length .

Finally, define a -coloring on so that any -size homogeneous subset would give an increasing or decreasing sequence of length .

**Problem 4**

Assume for a contradiction there is such a decomposition , , and translations . Choose much larger than . Argue that

differs from only by a constant number of elements (independent of ). Do the same for the , and use the fact that to derive that

which is impossible for large enough .