Cardinal characteristics and large continuum

Georg Cantor's "Continuum Hypothesis", the question about the size or "cardinality" of the real line, was the first on David Hilbert's famous list of 23 problems that he proposed in 1900: is the cardinality of the real line the next cardinality after the cardinality of the natural numbers, or are there subsets of the real line which are neither countable nor equinumerous with the real line itself?  This question naturally leads to the investigation of subsets (often quite pathological subsets) of the real line.

In this project we propose to study and develop techniques for building set-theoretic universes (that is, mathematical structures that satisfy the set-theoretic axioms ZFC) in which subsets of the real line with predescribed properties (typical example: not Lebesgue-measurable, but of small cardinality) exist.  The techniques considered are variants of the method of "iterated forcing"; we point out several issues that cannot be solved by the current methods and try develop new methods.