The yellow cake

Abstract:

In this paper we consider the following property: $(\circledast ^{\mathrm {Da}})$ For every function $f:\mathbb {R} \times \mathbb {R}\longrightarrow \mathbb {R}$ there are functions $g^0_n$, $g^1_n:\mathbb {R}\longrightarrow \mathbb {R}$ (for $n<\omega$) such that \[ (\forall x,y\in \mathbb {R})(f(x,y)=\sum _{n<\omega }g^0_n(x)g^1_n(y)).\] We show that, despite some expectation suggested by S. Shelah (1997), $(\circledast ^{\mathrm {Da}})$ does not imply $\mathbf {MA}(\sigma \mbox {-centered})$. Next, we introduce cardinal characteristics of the continuum responsible for the failure of $(\circledast ^{\mathrm {Da}})$.