Extremal Uniquely Resolvable Multisets

TL;DR

This paper investigates the maximum size of multisets of subsets of [m] with a unique partition property, providing bounds and exact values in different regimes, advancing understanding of extremal combinatorial structures.

Contribution

It establishes new lower and upper bounds for the size of such multisets and computes exact values in specific cases, improving extremal combinatorics knowledge.

Findings

Lower bound of g(n,m) as (rac{nm}{\u221a{ }})

Upper bound of g(n,m) as (rac{n}{c}\u221a{rac{1}{c}}) for n=2^{cm}

Exact computation of g(n,m) for n near 2^{m-1}

Abstract

For positive integers $n$ and $m$, consider a multiset of non-empty subsets of $[m]$ such that there is a \textit{unique} partition of these subsets into $n$ partitions of $[m]$. We study the maximum possible size $g(n,m)$ of such a multiset. We focus on the regime $n \leq 2^{m-1}-1$ and show that $g(n,m) \geq \Omega(\frac{nm}{\log_2 n})$. When $n = 2^{cm}$ for any $c \in (0,1)$, this lower bound simplifies to $\Omega(\frac{n}{c})$, and we show a matching upper bound $g(n,m) \leq O(\frac{n}{c}\log_2(\frac{1}{c}))$ that is optimal up to a factor of $\log_2(\frac{1}{c})$. We also compute $g(n,m)$ exactly when $n \geq 2^{m-1} - O(2^{\frac{m}{2}})$.