Maciej Korpalski: Straightening almost chains into barely altenating ones

Tuesday, October 24, 2023 17:00

Location: room 601, Mathematical Institute, University of Wroclaw

Speaker:
Maciej Korpalski

Title: Straightening almost chains into barely altenating ones

Abstract: Consider an almost chain \(\mathcal{A} = \{A_x \subset \omega: x \in X\}\) for some separable linearly ordered set \(X\). Such a chain is barely alternating if for all \(n \in \omega\) we cannot find elements \(x_1 < x_2 < x_3 < x_4\) in \(X\) satisfying \(n \in A_{x_1}, A_{x_3}\), \(n \notin A_{x_2}, A_{x_4}\). We will show that under \(MA(\kappa)\), if \(|X| \leq \kappa\), then we can straighten our almost chain \(\mathcal{A}\) into a barely alternating one by changing at most finitely many elements in each set \(A_x\).