Assaf Rinot: Generalizing Erdős-Rado to singular cardinals

Toronto Set Theory Seminar
Friday, February 3 from 1:30 to 3pm
Fields, Room 210

Speaker: Assaf Rinot (Fields Institute and UTM)

Title: Generalizing Erdős-Rado to singular cardinals

Abstract:

One of the most famous implications of the infinite Ramsey theorem (1929) asserts that any infinite poset either contains an infinite antichain or an infinite chain. Ramsey’s theorem has been generalized by Dushnik and Miller (1941), and subsequently by Erdős to a theorem that implies that any poset of uncountable cardinality k either contains an antichain of size k, or an infinite chain.

Is it possible to ask for a more sophisticated second alternative? More specifically, can the theorem be strengthened to yield the existence of an infinite chain *with a maximal element*? This question, restricted to uncountable regular cardinals, was answered by Erdos and Rado (1956).

In this talk, we shall discuss the missing case – singular cardinals – and present a proof of a Theorem of Shelah (2009) in the positive direction.

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