  We have observations concerning the set theoretic strength of the following
combinatorial statements without the axiom of choice. 1. If in a partially
ordered set, all chains are finite and all antichains are countable, then the
set is countable. 2. If in a partially ordered set, all chains are finite and
all antichains have size $\aleph_{\alpha}$, then the set has size
$\aleph_{\alpha}$ for any regular $\aleph_{\alpha}$. 3. CS (Every partially
ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF
(Every partially ordered set has a cofinal well-founded subset). 5. DT
(Dilworth's decomposition theorem for infinite p.o.sets of finite width). 6. If
the chromatic number of a graph $G_{1}$ is finite (say $k<\omega$), and the
chromatic number of another graph $G_{2}$ is infinite, then the chromatic
number of $G_{1}\times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G},
E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$
has a homomorphism into $H$, then so has $G$. Further we study a few statements
restricted to linearly-ordered structures without the axiom of choice.
