296 research outputs found
Chromatic number of the product of graphs, graph homomorphisms, Antichains and cofinal subsets of posets without AC
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 , then the set has size
for any regular . 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 is finite (say ), and the
chromatic number of another graph is infinite, then the chromatic
number of is . 7. For an infinite graph and a finite graph , if every finite subgraph of
has a homomorphism into , then so has . Further we study a few statements
restricted to linearly-ordered structures without the axiom of choice.Comment: Revised versio
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