The second author has recently shown ([20])that any selectively (a) almost disjoint family must have cardinality strictly less than 2א0 , so under the Continuum Hypothesis such a family is necessarily countable. However, it is also shown in the same paper that 2א0 < 2א1 alone does not avoid the existence of uncountable selectively (a) almost disjoint families. We show in this paper that a certain effective parametrized weak diamond principle is enough to ensure countability of the almost disjoint family in this context. We also discuss the deductive strength of this specific weak diamond principle (which is consistent with the negation of the Continuum Hypothesis, apart from other features). 

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HJM_M_dS_selectively_(a)_under_parametrized_diamond_2nd_revised_compiling_problems.pdf