Set families with a forbidden pattern

A balanced pattern of order 2d is an element P ∈ {+, −}2d , where both signs appear d times. Two sets A, B ⊂ [n] form a P-pattern, which we denote by pat(A, B) = P, if A△B = {j1, . . . , j2d} with 1 ≤ j1 < · · · < j2d ≤ n and {i ∈ [2d] : Pi = +} = {i ∈ [2d] : ji ∈ A \ B}. We say A ⊂ P...