Largest family without A union B contained in C intersect D
  Let F be a family of subsets of an n-element set not containing four distinct members such that A union B is contained in C intersect D. It is proved that the maximum size of F under this condition is equal to the sum of the two largest binomial coefficients of order n. The maximum families are also characterized. A LYM-type inequality for such families is given, too. 
Annalisa De Bonis, Gyula O.H. Katona, Konrad J. Swanepoel