Osvaldo Guzman Gonzalez: On weakly universal functions

Thursday, April 12, 2018, from 4 to 5:30pm
East Hall, room 3088

Speaker: Osvaldo Guzman Gonzalez (York University)

Title: On weakly universal functions

Abstract:

A function U:[omega_1]^2 —> 2 is called universal if for every function F:[omega_1]^2 —> omega there is an injective function h:omega_1 —> omega_1 such that F(alpha,beta)=U(h(alpha),h(beta)) for each alpha,betain omega_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 Steprans, where 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:[omega_1]^2 —> omega is (1,omega_1)-weakly universal if for every F:[omega_1]^2 —> omega there is an injective function h:omega_1 —> omega_1 and a function e:omega —> omega such that F(alpha,beta)=eU(h(alpha),h(beta)) for every alpha,betain omega_1. Shelah and Steprans asked if (1,omega_1)-weakly universal functions exist in ZFC. We will study the existence of (1,omega_1)-weakly universal functions in Sacks models and provide an answer to their problem.

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