Multiply union families in $\mathbb{N}^n$

TL;DR

This paper determines the maximum size and unique structure of certain families of non-negative integer sequences in high-dimensional spaces, characterized by an r-wise s-union property, for large dimensions or when the dimension equals r+1.

Contribution

It provides exact maximum sizes and characterizes the unique extremal configurations of r-wise s-union families in ^n for large n or when n=r+1.

Findings

Maximum size of r-wise s-union families in ^n is established.

Unique extremal configurations are characterized for large n or n=r+1.

Results extend understanding of union families in combinatorics.

Abstract

Let $A\subset \mathbb{N}^{n}$ be an $r$-wise $s$-union family, that is, a family of sequences with $n$ components of non-negative integers such that for any $r$ sequences in $A$ the total sum of the maximum of each component in those sequences is at most $s$. We determine the maximum size of $A$ and its unique extremal configuration provided (i) $n$ is sufficiently large for fixed $r$ and $s$, or (ii) $n=r+1$.