According to P. Erdõs, A. Hajnal: On a property of families of sets, Acta Math. Acad. Sci. Hungar. 12 (1961), 87--123 (see page 90) the stronger result you formulated is a theorem of Bernstein from F. Bernstein, "Zur Theorie der trigonometrischen Reihen", Leipz. Bet. (Berichte über die Verhandlungen der Königl. Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Math.phys. Klasse) 60 (1908), 325--338" MR. If you are looking for further results, "property B" is the keyword you need.
I presented a minicourse on some related properties of families of sets on the 7th Young Set Theory Workshop in 2014, so my slides may contain some results you are interested in.
