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Miklós Schweitzer 1986, Problem 1

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September 12, 2016
Miklos Schweitzercollege contestsset theory

Problem Statement

If (A,<)(A, <) is a partially ordered set, its dimension, dim⁡(A,<)\dim (A, <), is the least cardinal κ\kappa such that there exist κ\kappa total orderings {<α ⁣:α<κ}\{ <_{\alpha} \colon \alpha < \kappa \} on AA with <=∩α<κ<α<=\cap_{\alpha < \kappa} <_\alpha. Show that if dim⁡(A,<)>ℵ0\dim (A, <)>\aleph_0, then there exist disjoint A0,A1⊆AA_0, A_1\subseteq A with dim⁡(A0,<)\dim (A_0, <), dim⁡(A1,<)>ℵ0\dim (A_1, <)>\aleph_0. [D. Kelly, A. Hajnal, B. Weiss]
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