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Strongly almost disjoint familes, revisited

A. HajnalIstvan JuhászSaharon Shelah — 2000

Fundamenta Mathematicae

The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if A [ κ ] λ with |A|=κ is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ϱ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ϱ. Then there is a ϱ-closed forcing P such that, in V P , we have both GCH and M ( ϱ ( + ϱ + 1 ) , ϱ + , ϱ ) B [resp. M ( ϱ ( + ϱ + 1 ) , λ , ϱ ) B ( ϱ + ) for all λ ϱ ( + ϱ + 1 ) ] . These show that, consistently, the results of [EH] are sharp. The necessity...

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