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  1. ; ; (, Bulletin of the London Mathematical Society)
    Abstract We strengthen a result of Bagaria and Magidor (Trans. Amer. Math. Soc.366(2014), no. 4, 1857–1877)  about the relationship between large cardinals and torsion classes of abelian groups, and prove thattheMaximum Deconstructibilityprinciple introduced in Cox (J. Pure Appl. Algebra226(2022), no. 5) requires large cardinals; it sits, implication‐wise, between Vopěnka's Principle and the existence of an ‐strongly compact cardinal.While deconstructibility of a class of modules always implies the precovering property by Saorín and Šťovíček (Adv. Math.228(2011), no. 2, 968–1007), the concepts are (consistently) nonequivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits. 
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  2. ; ; (, Forum of Mathematics, Sigma)
    Abstract Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $$\mathcal {F}$$ of (real or complex) analytic functions, such that $$\big \{ f(x) \ : \ f \in \mathcal {F} \big \}$$ is countable for every x . We strengthen Erdős’ result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. We use such systems to construct, assuming CH, an equivalence relation $$\sim $$ on $$\mathbb {R}$$ such that any ‘analytic-anonymous’ attempt to predict the map $$x \mapsto [x]_\sim $$ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman [2]. 
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