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Speaker: Andras Zsak, Texas A&M University

Title: Half-filling families of finite sets

Abstract: (pdf version) Let F be a collection of finite subsets of a set X. We say that F is hereditary if B \in F whenever B \subset A \in F; we say F is half-filling if it is hereditary and for every finite set S \subset X there is a set A \in F such that A \subset S and |A| >= |S|/2. The following Ramsey-type problem is still open: given a half-filling family on \omega_1 (the first uncountable ordinal) is there an infinite subset of \omega_1 all whose finite subsets belong to F?

It can be shown that a negative answer would imply that for every countable ordinal \alpha there is a minimal compact half-filling family on N with Cantor-Bendixon index at least \alpha. Our main result is that this consequence of a negative answer is in fact true.

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