Set families with a forbidden pattern

TL;DR

This paper investigates the maximum size of set families avoiding certain balanced sign patterns in symmetric differences, establishing bounds that depend on the pattern's structure and size relative to n.

Contribution

It introduces new extremal bounds for P-free families in the Boolean lattice, especially for patterns with bounded balancedness and specific sign arrangements.

Findings

For fixed c>0, P-free families are o(2^n) when d < c log log n.

Stronger bounds are established for patterns with all pluses then minuses, and alternating signs, when d = o(√n).

Bounds are tight for patterns with all pluses followed by minuses.

Abstract

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. .