Multiple gaps and some finitizations of club and CH

We continue the development of the theory of capturing schemes over ${\omega }_1$ by analyzing the relation between the capturing construction schemes (whose existence is implied by Jensen’s $\mathrm{◊}$-principle) and both the Continuum Hypothesis and Ostaszewski’s $\mathrm{♣}$-principle. Formally, we show that the property of being capturing can be viewed as the conjunction of two properties, one of which is implied by $\mathrm{♣}$ and the other one by CH. We apply these principles to construct multiple gaps, entangled sets and metric spaces without uncountable monotone subspaces.