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Splitting Families, Reaping Families, and Families of Permutations Associated with Asymptotic Density

2025/06/26 by David Valderrama, Valderrama, David
Computer Science · Mathematics · #03E17 #03E35 #Advanced Graph Theory Research #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2506.21059

openalex publication_date 2025/06/26 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

We investigate several relations between cardinal characteristics of the continuum related with the asymptotic density of the natural numbers and some known cardinal invariants. Specifically, we study the cardinals of the form \mathfraksX, \mathfrakrX and \mathfrakddX,Y introduced in arXiv:2304.09698 and arXiv:2410.21102, answering some questions raised in these papers. In particular, we prove that \mathfraks0= cov(M) and \mathfrakr0= non(M). We also show that \mathfrakdd_\r\, \textsfall=\mathfrakdd_\1/2\, \textsfall for all r∈ (0,1), and we provide a proof of Con(\mathfrakdd_(0,1),\0,1\^\textsfrel< non(N)) and Con(\mathfrakdd_\textsfall,\textsfall^\textsfrel< non(N)).

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