# Relationship between “infinitely unequal” and “eventually different”

Suppose that $$\kappa$$ has the property that for every family $$A\subseteq\omega^{\omega}$$, if $$|A|<\kappa$$, then there exists some $$g\in\omega^{\omega}$$ such that for any $$f\in A, \exists^{\infty}n\;f(n)\neq g(n)$$. Does it then follow that for any family $$A\subseteq\omega^{\omega}$$ of size $$<\kappa$$, there exists a $$g\in\omega^{\omega}$$ such that for any $$f\in A, \forall^{\infty}n\;f(n)\neq g(n)$$?

It does not follow. In fact, it is not true in the Cohen model. Let $$X$$ be a set of $$\aleph_2$$ mutually generic Cohen reals over $$V$$. (Recall: mutually generic'' means that if $$x \in X$$ and $$Y \subseteq X$$, then $$x \in V[Y]$$ if and only if $$x \in Y$$, and otherwise $$x$$ is Cohen-generic over $$V[Y]$$.)
In $$V[X]$$, if $$A \subseteq \omega^\omega$$ with $$|A| < \aleph_2$$, then (because $$A$$ is an object that is hereditarily of size $$\leq\!\aleph_1$$) there is some $$X_0 \subseteq X$$ with $$|X_0| = \aleph_1$$ such that $$A \subseteq V[X_0]$$. If $$g \in X \setminus X_0$$, then $$g$$ is Cohen-generic over $$V[X_0]$$, and hence over any $$f \in A$$. In particular, for any $$f \in A$$, $$\exists^\infty n f(n) \neq g(n)$$.
On the other hand, let $$A \subseteq X$$ with $$|A| = \aleph_1$$. If $$g \in \omega^\omega$$, then (because $$g$$ is a hereditarily countable object) there is some countable $$A_0 \subseteq A$$ such that $$g \in V[A_0]$$. If $$f \in A \setminus A_0$$, then it is Cohen-generic over $$V[A_0]$$, and hence $$\exists^\infty n f(n) = g(n)$$.