Set families with a forbidden pattern

A balanced pattern of order 2d is an element P ∈ {+, −}^2d, whereboth 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, . . . , j_2d} with 1 ≤ j₁ < · · · < j_2d ≤ n and {i ∈ [2d] : P_i = +} = {i ∈ [2d] : j_i ∈ A \ B}.We say A ⊂ P [n] is P-free if pat(A, B) ≠ P for all A, B ∈ A. Weconsider the following extremal question: how large can a family A ⊂ P [n] be if A is P-free?

We prove a number of results on the sizes of such families.In particular, we show that for some fixed c > 0, if P is ad-balanced pattern with d < c log log n then |A|= o(2ⁿ). Wethen give stronger bounds in the cases when (i) P consists of d+signs, followed by d− signs and (ii) P consists of alternating signs.In both cases, if d = o(√n) then |A|= o(2ⁿ). In the case of (i), thisis tight.