The following statement is a direct consequence of the Continuum Hypothesis:

There exists a sequence $\langle f_\alpha:\omega_1\rightarrow\omega_1 ~ \vert ~ \alpha<\omega_1\rangle$ of functions such that there is no function $f:\omega_1\rightarrow\omega_1$ with the property that the sets $\{\xi<\omega_1 ~ \vert ~ f(\xi)=f_\alpha(\xi)\}$ are finite for all $\alpha<\omega_1$.

Moreover, since a failure of this statement can be used to obtain an $\omega_2$-sequence of subsets of $\omega_1$ with pairwise finite intersection, results of Baumgartner in

*Baumgartner, James E.*, **Almost-disjoint sets, the dense set problem and the partition calculus**, Ann. Math. Logic 9, 401-439 (1976). ZBL0339.04003.

show that the statement is not equivalent to CH.

**Question:** Can the above statement consistently fail?

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