If there is no inner model with measurable cardinals, then for each cardinal $\lambda $ there is an almost disjoint family $\Cal A_{\lambda}$ of countable subsets of $\lambda $ such that every subset of $\lambda $ with order type $\geq {\omega^{\scriptscriptstyle2}}$ contains an element of $\Cal A_{\lambda}$.
@article{116976,
author = {Lajos Soukup},
title = {On $\omega^2$-saturated families},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {32},
year = {1991},
pages = {355-359},
zbl = {0755.03027},
mrnumber = {1137796},
language = {en},
url = {http://dml.mathdoc.fr/item/116976}
}
Soukup, Lajos. On $\omega^2$-saturated families. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) pp. 355-359. http://gdmltest.u-ga.fr/item/116976/
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