{"paper":{"title":"Extremal Uniquely Resolvable Multisets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Varun Sivashankar","submitted_at":"2021-09-17T21:41:13Z","abstract_excerpt":"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)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.10222","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.10222/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}