Mathematica Eterna
Open Access

Abstract

Strong sequences and independent sets

Joanna Jureczko

A family S ∈ P(ω) is an independent family if for each pair A, B of disjoint finite subsets of S the set T A ∩ (ω \ S B) is nonempty. The fact that there is an independent family on ω of size continuum was proved by Fichtenholz and Kantorowicz in [7]. If we substitute P(ω) by a set (X, r) with arbitrary relation r it is natural question about existence and length of an independent set on (X, r). In this paper special assumptions of such existence will be considered. On the other hand in 60s’ of the last century the strong sequences method was introduced by Efimov. He used it for proving some famous theorems in dyadic spaces like: Marczewski theorem on cellularity, Shanin theorem on a calibre, Esenin-Volpin theorem and others. In this paper there will be considered: length of strong sequences, the length of independent sets and other well known cardinal invariants and there will be examined inequalities among them.