The topological pigeonhole principle for ordinals

Given a cardinal $\kappa$ and a sequence $\left(\alpha_i\right)_{i\in\kappa}$ of ordinals, we determine the least ordinal $\beta$ (when one exists) such that the topological partition relation $$\beta\rightarrow\left(top\,\alpha_i\right)^1_{i\in\kappa}$$ holds, including an independence result for one class of cases. Here the prefix "$top$" means that the homogeneous set must have the correct topology rather than the correct order type. The answer is linked to the non-topological pigeonhole principle of Milner and Rado.Comment: 24 page

Combinatorics of countable ordinal topologies

We study combinatorial properties of ordinals under the order topology, focusing on the subspaces, partition properties and autohomeomorphism groups of countable ordinals. Our main results concern topological partition relations. Let n be a positive integer, let κ be a cardinal, and write [X] n for the set of subsets of X of size n. Given an ordinal β and ordinals αi for all i ∈ κ, write β →top (αi) n i∈κ to mean that for every function c : [β] n → κ (a colouring) there is some subspace X ⊆ β and some i ∈ κ such that X is homeomorphic to αi and [X] n ⊆ c −1 ({i}). We examine the cases n = 1 and n = 2, defining the topological pigeonhole number P top (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 1 i∈κ , and the topological Ramsey number Rtop (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 2 i∈κ . We resolve the case n = 1 by determining the topological pigeonhole number of an arbitrary sequence of ordinals, including an independence result for one class of cases. In the case n = 2, we prove a topological version of the Erd˝os–Milner theorem, namely that Rtop (α, k) is countable whenever α is countable and k is finite. More precisely, we prove that Rtop(ω ω β , k + 1) ≤ ω ω β·k for all countable ordinals β and all positive integers k. We also provide more careful upper bounds for certain small ordinals, including Rtop(ω + 1, k + 1) = ω k + 1, Rtop(α, k) < ωω whenever α < ω2 , Rtop(ω 2 , k) ≤ ω ω and Rtop(ω 2 + 1, k + 2) ≤ ω ω·k + 1 for all positive integers k. Outside the partition calculus, we prove a topological analogue of Hausdorff’s theorem on scattered total orderings. This allows us to characterise countable subspaces of ordinals as the order topologies of countable scattered total orderings. As an application, we compute the number of subspaces of an ordinal up to homeomorphism. Finally, we study the group of autohomeomorphisms of ω n ·m+1 for finite n and m. We classify the normal subgroups contained in the pointwise stabiliser of the limit points. These subgroups fall naturally into D (n) disjoint sets, each either countable or of size 22 ℵ0 , where D (n) is the number of ⊆-antichains of P ({1, 2, . . . , n}). Our techniques span a variety of disciplines, including set theory, general topology and permutation group theor