Set families with a forbidden pattern

A balanced pattern of order $2d$ is an element $P\in \{+,-{\}}^{2d}$, where both signs appear $d$ times. Two sets $A,B\subset [n]$ form $P$-pattern, which we denote by $\mathrm{pat}(A,B)=P$, if $A\triangle B = {j_1,\ldots ,j_{2d}}$with$1\leq j_1<\cdots < j_{2d}\leq n$and${i\in [2d]: P_i = + } = {i\in [2d]: j_i \in A \setminus B}$\mathrm{.}Wesay${\cal A} \subset {\cal P}[n]$is$P$-freeif$\operatorname{pat}(A,B)\neq P$forall$A,B \in {\cal A}$. We consider the following extremal question: how large can a family ${\cal A} \subset {\cal P}[n]$beif${\cal 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 a $d$-balanced pattern with $d < c \log \log n$then$|{\cal A}| = o(2^n)$. We then 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 $d=o(\sqrt{n})$ then $\mathrm{\mid }\mathcal{A}\mathrm{\mid }=o(2^n)$. In the case of (i), this is tight. .