The erdös-dushnik-miller theorem for topological graphs and orders

摘要

A topological graph is a graphG=(V, E)on a topological spaceVsuch that the edge setEis a closed subset of the product spaceV x V. If the graph contains no infinite independent set then, by a well-known theorem of Erdös, Dushnik and Miller, for any infinite setL⊑V, there is a subsetL′⊑Lof the same oardinality L′ = L such that the restrictionG↾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 subsetA⊑Vdoes not contain an infinite independent set. If the cofinality cf (L)>A, then there is anL′ as before, but if cf (L)ω, there is a subsetL′⊑Lof size L′=L such thatG↾L′ is complete. The