{"paper":{"title":"Asymptotically optimal lower bounds on weak saturation numbers for hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Polymatroids establish asymptotically optimal lower bounds on weak saturation numbers for hypergraphs.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nikolai Terekhov","submitted_at":"2026-04-08T13:56:51Z","abstract_excerpt":"Given an $r$-uniform hypergraph $H$ and a positive integer $n$, the weak saturation number $\\mathrm{wsat}(n,H)$ is the minimum number of edges in an $r$-uniform hypergraph $F$ on $n$ vertices such that the missing edges in $F$ can be added, one at a time, so that each added edge creates a copy of $H$.\n  For the case of graphs ($r = 2$), asymptotically optimal general lower bounds for these numbers in terms of the minimum vertex degree of $H$ are known. In this work, we generalize these bounds to the case of hypergraphs and establish their asymptotic optimality. To prove this, we introduce a lo"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We generalize these bounds to the case of hypergraphs and establish their asymptotic optimality. To prove this, we introduce a lower bound method based on polymatroids.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The polymatroid construction correctly captures the weak saturation process for arbitrary r-uniform H and yields tight asymptotic coefficients without hidden integrality constraints.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalizes asymptotically optimal lower bounds on weak saturation numbers wsat(n,H) from graphs to r-uniform hypergraphs via a polymatroid method that yields non-integer coefficients.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Polymatroids establish asymptotically optimal lower bounds on weak saturation numbers for hypergraphs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"94f555ec0546cad7b255a7af8b40c94391af522255eaf624863539047f67c6f4"},"source":{"id":"2604.07104","kind":"arxiv","version":2},"verdict":{"id":"cc11edaa-9e2a-4cec-b450-a58b44b28d51","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T17:33:40.570972Z","strongest_claim":"We generalize these bounds to the case of hypergraphs and establish their asymptotic optimality. To prove this, we introduce a lower bound method based on polymatroids.","one_line_summary":"Generalizes asymptotically optimal lower bounds on weak saturation numbers wsat(n,H) from graphs to r-uniform hypergraphs via a polymatroid method that yields non-integer coefficients.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The polymatroid construction correctly captures the weak saturation process for arbitrary r-uniform H and yields tight asymptotic coefficients without hidden integrality constraints.","pith_extraction_headline":"Polymatroids establish asymptotically optimal lower bounds on weak saturation numbers for hypergraphs."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.07104/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"}