Topological Ramsey numbers and countable ordinals

Por • 30 oct, 2015 • Sección: Educacion

Andrés Eduardo Caicedo, Jacob Hilton

Abstract: We study the topological version of the partition calculus in the setting of countable ordinals. Let α and β be ordinals and let k be a positive integer. We write βtop(α,k)2 to mean that, for every red-blue coloring of the collection of 2-sized subsets of β , there is either a red-homogeneous set homeomorphic to α or a blue-homogeneous set of size k . The least such β is the topological Ramsey number Rtop(α,k) . We prove a topological version of the Erdos-Milner theorem, namely that Rtop(α,k) is countable whenever α is countable. More precisely, we prove that Rtop(ωωβ,k+1)≤ωωβk for all countable ordinals β and finite k . Our proof is modeled on a new easy proof of a weak version of the Erdos-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of α , proving among other results that Rtop(ω+1,k+1)=ωk+1 , Rtop(α,k)<ωω whenever α<ω2 , Rtop(ω2,k)≤ωω and Rtop(ω2+1,k+2)≤ωωk+1 for all finite k . Our computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments.

arXiv:1510.00078v1 [math.LO]

Logic (math.LO)

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