Intersection properties of open sets, II.
  A topological space is called P_2 ( P_3, P_{<omega} ) if and only if it does not contain two (three, finitely many) uncountable open sets with empty intersection.   We show that   (i) there are 0-dimensional P_{<omega} spaces of size 2^omega,   (ii) there are compact P_{<omega} spaces of size omega_1,   (iii) the existence of a Psi-like examples for a compact P_{<omega} space of size omega_1 is independent of ZFC,   (iv) it is consistent that 2^omega is as large as you wish but every first countable (and so every compact) P_2 space has cardinality<=omega_1. 
I. Juh\'asz, Zs. Nagy, Lajos Soukup, Z. Szentmikl\'ossy