The Borel conjecture (Q753806)

A subset X of \({\mathbb{R}}\) has strong measure zero if for every sequence \(\epsilon_ 0,\epsilon_ 1,\epsilon_ 2,..\). of positive reals there exists a sequence \(I_ 0,I_ 1,I_ 2,..\). of open intervals such that, for each \(i<\omega\), \(length(I_ i)<\epsilon_ i\) and \(X\subseteq \cup_{i\in \omega}I_ i.\) In 1919, E. Borel conjectured that every strong measure zero subset of \({\mathbb{R}}\) is (at most) countable. In 1928, W. Sierpinski remarked that the set constructed by N. Lusin in 1914 on the assumption of the continuum hypothesis (an uncountable subset of \({\mathbb{R}}\) having countable intersection with each first category set) has strong measure zero. In 1976, R. Laver showed that Borel's conjecture is consistent, if Zermelo- Fraenkel set theory is consistent. In the model he obtained for forcing, the cardinality of \({\mathbb{R}}\) is \(\aleph_ 2.\) The authors show that Borel's conjecture is also consistent with assumptions saying that the continuum is ``large''. This requires a forcing technique different from the ``countable support iteration technique'' used by Laver. The authors show that ``adding random reals'' to the Laver model preserves the Borel conjecture. Their result improves earlier unpublished results by the third author.