Tag Archives: Almost-disjoint family

Theorem (Engelking-Karlowicz, 1965). For cardinals $\kappa \le \lambda \le \mu \le 2^{\lambda }$, the following are equivalent: ${\lambda }^{<\kappa }=\lambda$; there exists a collection of functions, $⟨f_i:\mu \to \lambda \mid i<\lambda ⟩$, such that for every $X\in [\mu {]}^{<\kappa }$ and every function $f:X\to \lambda$, there exists some $i<\lambda$ with $f\subseteq f_i$. Proof. (2)$\Rightarrow$(1) Suppose … Continue reading →