We give bad (with respect to the reverse inclusion ordering) sequences of monomial ideals in two variables with Ackermannian lengths and extend this to multiple recursive lengths for more variables.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm216-2-1,
author = {Florian Pelupessy and Andreas Weiermann},
title = {On the lengths of bad sequences of monomial ideals over polynomial rings},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {101-108},
zbl = {1244.03122},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm216-2-1}
}
Florian Pelupessy; Andreas Weiermann. On the lengths of bad sequences of monomial ideals over polynomial rings. Fundamenta Mathematicae, Tome 219 (2012) pp. 101-108. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm216-2-1/