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 $\operatorname{pat}(A,B) = P$, if $A\triangle B = \{j_1,\ldots ,j_{2d}\}$ with $1\leq j_1<\cdots < j_{2d}\leq n$ and $\{i\in [2d]: P_i = + \} = \{i\in [2d]: j_i \in A \setminus B\}$. We say ${\cal A} \subset {\cal P}[n]$ is $P$-free if $\operatorname{pat}(A,B)\neq P$ for all $A,B \in {\cal A}$. We consider the following extremal question: how large can a family ${\cal A} \subset {\cal P}[n]$ be if ${\cal A}$ 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 $|{\cal A} | = o(2^n)$. In the case of (i), this is tight. .

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