slideshow 3

Seminar on Reckoning

On weakly universal functions

Osvaldo Guzman
University of Toronto


Seminar on Reckoning
Wednesday, 21. August 2019 - 11:00 to 15:00

Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building

A function U: [ω_1]^2 -> 2 is called universal if for every function F: [ω_1]^2 -> ω there is an injective function h: ω_1 -> ω_1 such that F(α,β) =U(h(α), h(β)) for each α,β ∈ ω_1. It is easy to see that universal functions exist assuming the Continuum Hypothesis, furthermore, by results of Shelah and Mekler, the existence of such functions is consistent with the continuum being arbitrarily large. Universal functions were recently studied by Shelah and Steprāns, they showed that the existence of universal graphs is consistent with several values of the dominating and unbounded numbers. They also considered several variations of universal functions, in particular, the following notion was studied: A function U: [ω_1]^2 -> ω is (1, ω_1)-weakly universal if for every F: [ω_1]^2-> ω there is an injective function h: ω_1 -> ω_1 and a function e: ω -> ω such that F(α,β) = e(U(h(α), h(β))) for every α,β ∈ ω_1. Shelah and Steprāns asked if (1, ω_1)-weakly universals functions exist in ZFC. We will study the existence of (1, ω_1)-weakly universal functions in Sacks models and provide an answer to their problem.