Problem 1

This is the same argument as in class. Just replace ${\{0,1\}^{\mathbb N}}$ by ${\{0,1\}^\kappa}$.

Problem 2

Again, generalize the argument from class. Consider the lexicographic ordering on ${\{0,1\}^\kappa}$, i.e. ${f < g}$ iff ${f(\alpha) < g(\alpha)}$ where ${\alpha}$ is the least ordinal such that ${f(\alpha) \neq g(\alpha)}$. This is a linear order on ${\{0,1\}^\kappa}$.

Argue that ${\{0,1\}^\kappa}$ with this ordering has no infinite decreasing or increasing sequence of length ${\kappa^+}$. For this, you have to “lift” the Pigeonhole Principle analysis to binary sequences of length ${\kappa}$.

Finally, define a ${2}$-coloring on ${2^\kappa = |\{0,1\}^\kappa|}$ so that any ${\kappa^+}$-size homogeneous subset would give an increasing or decreasing sequence of length ${\kappa^+}$.

Problem 4

Assume for a contradiction there is such a decomposition ${A = \bigcup A_i}$, ${B = \bigcup B_j}$, and translations ${a_1, \ldots, a_m}$ ${b_1, \ldots, b_n}$. Choose ${N}$ much larger than ${m,n,a_1, \ldots, a_m, b_1, \ldots, b_n}$. Argue that

$\displaystyle \bigcup (A_i \cap [-N,N]) + a_i$

differs from ${[-N,N]}$ only by a constant number of elements (independent of ${N}$). Do the same for the ${B_i}$, and use the fact that ${{\mathbb Z} = A \cup B}$ to derive that

$\displaystyle 2 |[-N,N]| \leq |[-N,N]| + \text{constant},$

which is impossible for large enough ${N}$.