## Combinatorial cardinal characteristics of the continuum.(English)Zbl 1198.03058

Foreman, Matthew (ed.) et al., Handbook of set theory. In 3 volumes. Dordrecht: Springer (ISBN 978-1-4020-4843-2/hbk; 978-1-4020-5764-9/ebook). 395-489 (2010).
Summary: The combinatorial study of subsets of the set $$N$$ of natural numbers and of functions from $$N$$ to $$N$$ leads to numerous cardinal numbers, uncountable but no larger than the continuum. For example, how many infinite subsets $$X$$ of $$N$$ must I take so that every subset $$Y$$ of $$N$$ or its complement includes one of my $$X$$’s? Or how many functions $$f$$ from $$N$$ to $$N$$ must I take so that every function from $$N$$ to $$N$$ is majorized by one of my $$f$$’s? The main results about these cardinal characteristics of the continuum are of two sorts: inequalities involving two (or sometimes three) characteristics, and independence results saying that other such inequalities cannot be proved in ZFC. Other results concern, for example, the cofinalities of these cardinals or connections with other areas of mathematics. This survey concentrates on the combinatorial set-theoretic aspects of the theory.