Extremal Uniquely Resolvable Multisets
| Autor: | Sivashankar, Varun |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $n \geq 2^{m-1} - O(2^{\frac{m}{2}})$. Comment: 12 pages, 4 figures. Accepted to SIAM Journal on Discrete Mathematics (SIDMA) |
| Databáze: | arXiv |
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