A weak $λ$-set that is not a $λ$-set under $\frak{p} = \frak{c}$

Abstract

Under $\mathfrak p=\mathfrak c$, Cantor space contains a weak $λ$-set that is not a $λ$-set. This answers the consistency question of de Oliveira Rodrigues, dos Santos Ronchim, and Szeptycki. The hypothesis $\mathfrak p=\mathfrak c$ is compatible with the failure of the continuum hypothesis. The construction combines the finite-unions form of Hindman's theorem with Bell's characterization of $\mathfrak p$ in terms of $σ$-centered forcing. The increasing enumerations of the infinite members of the constructed space form an unbounded family under eventual domination. This makes the countable set of finite subsets of $ω$ a witness to the failure of the $λ$-property.