Sparse analytic systems

Abstract

Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal {F}$ of (real or complex) analytic functions, such that $\big \{ f(x) \ : \ f \in \mathcal {F} \big \}$ is countable for every x. We strengthen Erdős’ result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. We use such systems to construct, assuming CH, an equivalence relation $\sim $ on $\mathbb {R}$ such that any ‘analytic-anonymous’ attempt to predict the map $x \mapsto [x]_\sim $ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman [2].

Wetzel families and the continuum

“…In Section 4, we show how to force a certain family of strongly almost disjoint functions that serves as a basic (and somewhat necessary) ingredient in the proof of the main result. In fact, † We have been informed that the authors of [11] have been unaware of [22], and were motivated rather by Erdös' original paper.…”

“…It seems that after [22], interest in Wetzel's problem has grown. We are aware of two more papers dealing with it since then, a formalisation of Erdős' proof, [26], and a proof that the continuum hypothesis implies the existence of sparse analytic systems, [11]. But no one yet addressed the open question which Kumar and Shelah ask at the end of [22], of whether the existence of a Wetzel family is consistent with a continuum of cardinality $$\aleph _2$$.…”