Reviewed Works:B. I. Zil'ber, L. Pacholski, J. Wierzejewski, A. J. Wilkie, Totally Categorical Theories: Structural Properties and the Non-Finite Axiomatizability.B. I. Zil'ber, Strongly Minimal Countably Categorical Theories.B. I. Zil'ber, Strongly Minimal Countably Categorical Theories. II.B. I. Zil'ber, Strongly Minimal Countably Categorical Theories. III.B. I. Zil'ber, E. Mendelson, Totally Categorical Structures and Combinatorial Geometries.B. I. Zil'ber, The Structure of Models of Uncountably Categorical Theories.

In this paper, we study the relationship between the cofinalityc(Sym(ω)) of the infinite symmetric group and the minimal cardinality $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{b} $$ of an unbounded familyF of ω ω.