A Galvin-Hajnal theorem for generalized cardinal characteristics

TL;DR

This paper extends classical theorems like Galvin-Hajnal and Silver's to a broad class of generalized cardinal characteristics at singular uncountable cofinality cardinals, revealing new structural properties.

Contribution

It establishes that various generalized cardinal characteristics satisfy analogues of the Galvin-Hajnal and Silver theorems at singular uncountable cofinality cardinals.

Findings

Generalized cardinal characteristics satisfy Galvin-Hajnal-type theorems

Results apply to meeting, reaping, and dominating numbers

Extends classical theorems to uncountable cofinality singular cardinals

Abstract

We prove that a variety of generalized cardinal characteristics, including meeting numbers, the reaping number, and the dominating number, satisfy an analogue of the Galvin-Hajnal theorem, and hence also of Silver's theorem, at singular cardinals of uncountable cofinality.