A note on tilted Sperner families with patterns

Let pp and qq be two nonnegative integers with p+q>0p+q>0 and n>0n>0. We call F⊂P([n])F⊂P([n]) a (p, q)-tilted Sperner family with patterns   on [n][n] if there are no distinct F,G∈FF,G∈F with: (i)p|F∖G|=q|G∖F|,and(ii)f>gfor allf∈F∖Gandg∈G∖F. E. Long in Long (2015) proved that the cardinality of a (1, 2)-tilted Sperner family with patterns on [n][n] is O(e120logn2nn). We improve and generalize this result, and prove that the cardinality of every (p,qp,q)-tilted Sperner family with patterns on [nn] is O(logn2nn).