Partitioning pairs of sigma-scattered linear orders, part 1

Seminar

Speaker

Thilo Weinert (BGU)

Date

20/04/2017 - 12:00 - 10:00Add to Calendar 2017-04-20 10:00:00 2017-04-20 12:00:00 Partitioning pairs of sigma-scattered linear orders, part 1 We are going to continue the analysis of generalised scattered orders, proving the result described towards the end of Chris Lambie-Hanson’s talk. This states that consistently, for every sigma-scattered linear ordering there is a colouring of its pairs in black & white such that every triple contains a white pair and every copy of one of the following order-types contains a black pair: omega_1^omega (omega_1^omega)^* omega_1 * (omega * omega^*)^omega omega_1^* * (omega * omega^*)^omega (omega * omega^*)^omega * omega_1 (omega * omega^*)^omega * omega_1^* This generalises a 46-year-old Theorem of Erdős & Rado about ordinals. A sufficient hypothesis implying this theorem is the existence of a colouring of the pairs of omega_1 * omega in black & white such that every triple contains a black pair and every subset of full order-type contains a white one. Time permitting we may present a proof that stick = b = Aleph_1 implies the existence of such a colouring. Here b is the unbounding number and stick = Aleph_1 is a weakening of the club principle which was considered by Baumgartner 41 years ago, named by Broverman, Ginsburg, Kunen & Tall two years thereafter and twenty years ago reconsidered as a cardinal characteristic by Fuchino, Shelah & Soukup. Building 604, Room 103 אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public

Place

Building 604, Room 103

Abstract

We are going to continue the analysis of generalised scattered orders, proving the result described towards the end of Chris Lambie-Hanson’s talk. This states that consistently, for every sigma-scattered linear ordering there is a colouring of its pairs in black & white such that every triple contains a white pair and every copy of one of the following order-types contains a black pair:

omega_1^omega
(omega_1^omega)^*
omega_1 * (omega * omega^*)^omega
omega_1^* * (omega * omega^*)^omega
(omega * omega^*)^omega * omega_1
(omega * omega^*)^omega * omega_1^*

This generalises a 46-year-old Theorem of Erdős & Rado about ordinals. A sufficient hypothesis implying this theorem is the existence of a colouring of the pairs of omega_1 * omega in black & white such that every triple contains a black pair and every subset of full order-type contains a white one. Time permitting we may present a proof that stick = b = Aleph_1 implies the existence of such a colouring. Here b is the unbounding number and stick = Aleph_1 is a weakening of the club principle which was considered by Baumgartner 41 years ago, named by Broverman, Ginsburg, Kunen & Tall two years thereafter and twenty years ago reconsidered as a cardinal characteristic by Fuchino, Shelah & Soukup.

Last Updated Date : 28/10/2019