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The Erdős-Dushnik-Miller theorem for topological graphs and orders. (English) Zbl 0574.05045
A topological graph is a graph $$G=(V,E)$$ on a topological space V such that the edge set E is a closed subset of the product space $$V\times V$$. If the graph contains no infinite independent set then, by a well-known theorem of Erdős, Dushnik and Miller, for any infinite set $$L\subseteq V$$, there is a subset L’$$\subseteq L$$ of the same cardinality $$| L'| =| L|$$ such that the restriction $$G\upharpoonright L'$$ is a complete graph. We investigate the question of whether the same conclusion holds if we weaken the hypothesis and assume only that some dense subset $$A\subseteq V$$ does not contain an infinite independent set. If the cofinality $$cf(| L|)>| A|$$, then there is an L’ as before, but if cf($$| L|)\leq | A|$$, then some additional hypothesis seems to be required. We prove that, if the graph $$G\upharpoonright A$$ is a comparability graph and A is a dense subset, then for any set $$L\subseteq V$$ such that $$cf(| L|)>\omega$$, there is a subset L’$$\subseteq L$$ of size $$| L'| =| L|$$ such that $$G\upharpoonright L'$$ is complete. The condition $$cf(| L| >\omega$$ is needed.

##### MSC:
 05C99 Graph theory 05A05 Permutations, words, matrices 03E05 Other combinatorial set theory
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