A note on tilted Sperner families with patterns

Let p and q be two nonnegative integers with p + q 0 and n 0 . We call F ź P ( n ) a (p, q)-tilted Sperner family with patterns on n if there are no distinct F , G ź F with: (i) p | F ź G | = q | G ź F | , and (ii) f g forźall f ź F ź G and g ź G ź F . E. Long in Long (2015) proved that the cardinality of a (1, 2)-tilted Sperner family with patterns on n is O ( e 120 log n 2 n n ) . We improve and generalize this result, and prove that the cardinality of every ( p , q )-tilted Sperner family with patterns on n is O ( log n 2 n n ) .