A solution to the problem of the existence of a dense plastic subset $X\subseteq\mathbb{R}$ of cardinality $|X|<\frak{c}$

TL;DR

The paper constructs a dense plastic subset of the real numbers with smaller cardinality than the continuum under certain set-theoretic assumptions, addressing a previously posed problem.

Contribution

It provides a specific example of such a subset under the assumption that the continuum is at least , solving an open problem.

Findings

Existence of a dense plastic subset with cardinality less than the continuum.

Construction relies on the assumption .

Answers a previously open problem in the field.

Abstract

Under the assumption $\frak{c}\geqslant \omega_2$, we give an example of a dense plastic subset $X\subseteq\mathbb{R}$ of cardinality $|X|<\frak{c}$. This answers Problem 1 of arXiv:2510.10537.