Strongly almost disjoint functions
A family of functions from \( \omega_1 \) to \( \omega \) is \textsl{strongly almost disjoint} if for any distinct \( f \) and \( g \) in the family, there are only finitely many \( \alpha \in \omega_1 \) such that \( f(\alpha) = g(\alpha) \). It is known that: (1) There is a strongly almost disjoint family of cardinality \( \aleph_1 \); and (2) The negation of Chang's Conjecture is equivalent to the existence of a c.c.c.\ poset that forces the existence of a strongly disjoint family of cardinality \( \aleph_2 \) [\textit{S. Todorcevic}, Can. J. Math. 43, 832-851 (1991; Zbl 0776.03024)]. The author considers the possible existence of even larger strongly almost disjoint families, and shows that for any cardinal \( \kappa \), there is a cardinal-preserving forcing notion that forces the existence of a strongly almost disjoint family of cardinality \( \kappa \). He assumes the Continuum Hypothesis in the ground model, and uses the side condition method due to \textit{S. Todorcevic} [Partition problems in topology (Contemp. Math. 84) (1989; Zbl 0659.54001)]. He remarks that it is also possible to obtain the same result by iterated forcing, and that this approach can be generalized to give a consistency proof of the existence of an arbitrarily large strongly almost disjoint family of functions from \( \omega_{n+1} \) to \( \omega_n \) (\( n \in \omega \)).
scientific article; zbMATH DE number 1011391
scientific article