For any partition of a set of squarefree numbers with relative density greater than 3/4 into two parts, at least one part contains three numbers whose product is a square. Also generalizations to partitions into more than two parts are discussed.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm120-1-1,
author = {M. Ska\l ba},
title = {Product of three numbers being a square as a Ramsey property},
journal = {Colloquium Mathematicae},
volume = {120},
year = {2010},
pages = {1-5},
zbl = {1280.11007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm120-1-1}
}
M. Skałba. Product of three numbers being a square as a Ramsey property. Colloquium Mathematicae, Tome 120 (2010) pp. 1-5. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm120-1-1/