Research toward a solution of Galvin's conjecture

2015 Fiscal Year Final Research Report

• Project/Area Number: 26610040
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kobe University
• Principal Investigator: Fuchino Sakae 神戸大学, システム情報学研究科, 教授 (30292098)
• Co-Investigator(Renkei-kenkyūsha): SAKAI Hiroshi 神戸大学, システム情報学研究科, 准教授 (70468239) ; USUBA Toshimichi 神戸大学, 自然科学系先端融合研究環, 助教 (10513632)
• Project Period (FY): 2014-04-01 – 2016-03-31
• Keywords: 反映原理 / 無限ゲーム / 彩色数 / 強コンパクト基数 / ω1-強コンパクト基数 / 巨大基数 / レヴィ崩壊

Outline of Final Research Achievements

Galvin's Conjecture is the assertion "any partial ordering X such that any subordering of X of size ω_1 is a union of countably many chains is by itself a union of countably many chains". Its consistency is still open. In our research, we introduced the reflection numbers corresponding to Rado's conjecture, Galvin's conjecture etc. and studied the relationships of between these cardinal numbers. Galvin's Conjecture is characterized by the corresponding reflection number being ω_2.
We showed the consistency of a restricted form of Galvin's conjecture claiming that the Galvin type reflection number can be ω_2 for the class of partial ordering for which the property that the partial ordering is not a union of countably many chains is preserved by σ-closed forcing and that for this class it is also consistent that the reflection number is less than or equal to the continuum while the continuum is fairly large.

Free Research Field

数理論理学，公理的集合論