Jonathan Schilhan (University of Leeds)

Wetzel's problem and the continuum

In the early 60's, John Wetzel came up with the following question in his PhD thesis on harmonic functions: If $\mathcal{F}$ is a family of entire functions (functions that are holomorphic on the complex plane) which at each point attains at most countably many values, is $\mathcal{F}$ itself necessarily countable? This question makes sense considering the quite restrictive nature of holomorphic functions. Not much thereafter, Erdős could show that a negative answer to Wetzel's Problem is in fact equivalent to the continuum hypothesis. His argument shows that any family of entire functions, that attains at each point less values than elements of that family, must have size continuum. Recently Kumar and Shelah have shown that consistently such a family exists while the continuum has size $\aleph_{\omega_1}$. We answer their main open problem by showing that continuum $\aleph_2$ is possible as well. This is joint work with Thilo Weinert.