On a generalization of a result of Kleitman

TL;DR

This paper generalizes a classical combinatorial result by determining the asymptotic maximum size of set families avoiding certain disjoint union configurations, extending previous work to more complex structures.

Contribution

It introduces a new generalized problem of set family maximum sizes avoiding specific disjoint union patterns, and provides asymptotic results for different modular conditions and parameters.

Findings

Asymptotic formulas for maximum set family sizes under new disjoint union constraints.

Different asymptotic behaviors identified for cases when n mod s equals -1 or 0.

Distinct results for the case t=2 compared to t=1 and t≥3.

Abstract

A classical result of Kleitman determines the maximum number $f(n,s)$ of subsets in a family $\mathcal{F}\subseteq 2^{[n]}$ of sets that do not contain distinct sets $F_1,F_2,\dots,F_s$ that are pairwise disjoint in the case $n\equiv 0,-1$ (mod $s$). Katona and Nagy determined the maximum size of a family of subsets of an $n$-element set that does not contain $A_1,A_2,\dots,A_t,B_1,B_2,\dots,B_t$ with $\bigcup_{i=1}^t A_i$ and $\bigcup_{i=1}^t B_i$ being disjoint. In this paper, we consider the problem of finding the maximum number $vex(n,K_{s\times t})$ in a family $\mathcal{F}\subseteq 2^{[n]}$ without sets $F^1_1,\dots,F^1_t,\dots,F^s_1,\dots,F^s_t$ such that $G_j=\bigcup_{i=1}^tF^j_i$ $j=1,2,\dots,s$ are pairwise disjoint. We determine the asymptotics of $2^n-vex(n,K_{s\times t})$ if $n\equiv -1$ (mod $s$) for all $t$, and if $n\equiv 0$ (mod $s$), $t\ge 3$ and show that in this latter case the asymptotics of the $t=2$ subcase is different from both the $t=1$ and $t\ge 3$ subcases.