d=ω1 implies a=ω1
Abstract
We prove in ZFC that d=ω1 implies a=ω1, settling a question of Roitman from the 1970s. Here d is the least size of a family that eventually dominates every function in ωω, and a is the least size of an infinite maximal almost disjoint (MAD) family of infinite subsets of ω. Consequently, a≤d whenever the continuum c=2ℵ0 is at most ω2. This answers negatively Shelah's question whether d<a is consistent with c=ω2, and shows that his model of d=ω2<a=c=ω3 has the least possible values of d, a and c among models of d<a. The proof also gives a preservation theorem: from any dominating family of size ω1, we construct a MAD family of size ω1 that remains maximal in every outer model in which the given family is still dominating. Thus, under d=ω1, some MAD family is indestructible by every ωω-bounding forcing; it can moreover be chosen almost strongly separable, hence Cohen-indestructible. These results give negative answers, under d=ω1, to two questions of Hrušák and to one of Brendle, Guzmán, Hrušák and Raghavan.
BibTeX
@misc{gomez2026dω1,
title = {{$\mathfrakd=ω_1$ implies $\mathfraka=ω_1$}},
author = {José de Jesús Pelayo Gómez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00639},
eprint = {2610.00639},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00639},
}
RIS
TY - UNPB
TI - d=ω_1 implies a=ω_1
AU - José de Jesús Pelayo Gómez
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00639
UR - https://arxiv.org/abs/2610.00639
AB - We prove in ZFC that d=ω_1 implies a=ω_1, settling a question of Roitman from the 1970s. Here d is the least size of a family that eventually dominates every function in ω^ω, and a is the least size of an infinite maximal almost disjoint (MAD) family of infinite subsets of ω. Consequently, a≤d whenever the continuum c=2^ℵ_0 is at most ω_2. This answers negatively Shelah's question whether d<a is consistent with c=ω_2, and shows that his model of d=ω_2<a=c=ω_3 has the least possible values of d, a and c among models of d<a. The proof also gives a preservation theorem: from any dominating family of size ω_1, we construct a MAD family of size ω_1 that remains maximal in every outer model in which the given family is still dominating. Thus, under d=ω_1, some MAD family is indestructible by every ω^ω-bounding forcing; it can moreover be chosen almost strongly separable, hence Cohen-indestructible. These results give negative answers, under d=ω_1, to two questions of Hrušák and to one of Brendle, Guzmán, Hrušák and Raghavan.
ER -