{"paper":{"title":"Shadows of Uniform Hypergraphs under a Minimum Degree Condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For vertex sets larger than a quadratic bound in t, any extremal k-uniform hypergraph with minimum degree binom(t, k-1) contains an isolated copy of K^k_{t+1}.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haorui Liu, Mei Lu, Yi Zhang","submitted_at":"2026-05-04T13:58:17Z","abstract_excerpt":"Given a set $X$ and an integer $t$, let $\\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal--Katona theorem implies that if $|\\mathcal{F}|\\geq \\binom{t}{k}$, then $|\\partial_\\ell\\mathcal{F}|\\geq\\binom{t}{\\ell}$. The minimum degree version of this problem asks: if $\\delta(\\mathcal{F})\\geq \\binom{t}{k-1}$, how small can $|\\partial_\\ell\\mathcal{F}|$ be? We call a hypergraph \\textit{extremal} if it achieves the minimum value of $|\\partial_\\ell \\mathcal{F}|$ subject to the degree condition $\\delta(\\mathcal{F}) \\geq \\binom{t}{k-1}$. F\\\"uredi and Zhao [SIAM J. Discrete Math. 36(4), 2022] pro"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By developing a hypergraph transformation that combines shifting operations with antilexicographic compression, we prove that there exists an extremal hypergraph containing an isolated copy of K^k_{t+1} whenever |X| > (1/4)(t+1)^2 binom(t-1, ℓ-2) + 2t.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The new shifting-plus-antilexicographic-compression transformation preserves both the minimum-degree condition and the extremal property for arbitrary k > ℓ ≥ 2.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For k-uniform hypergraphs with δ(F) ≥ binom(t, k-1), the minimum |∂_ℓ F| is achieved by one containing an isolated K_{t+1}^k when |X| > (1/4)(t+1)^2 binom(t-1, ℓ-2) + 2t.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For vertex sets larger than a quadratic bound in t, any extremal k-uniform hypergraph with minimum degree binom(t, k-1) contains an isolated copy of K^k_{t+1}.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d6131bdf1d2b4ed08e314b18abdf8ccc5af9828bfa44c0678308debd34c91431"},"source":{"id":"2605.02610","kind":"arxiv","version":2},"verdict":{"id":"375938ae-2e0f-4bae-8c52-c8211b0147a0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T17:45:04.664185Z","strongest_claim":"By developing a hypergraph transformation that combines shifting operations with antilexicographic compression, we prove that there exists an extremal hypergraph containing an isolated copy of K^k_{t+1} whenever |X| > (1/4)(t+1)^2 binom(t-1, ℓ-2) + 2t.","one_line_summary":"For k-uniform hypergraphs with δ(F) ≥ binom(t, k-1), the minimum |∂_ℓ F| is achieved by one containing an isolated K_{t+1}^k when |X| > (1/4)(t+1)^2 binom(t-1, ℓ-2) + 2t.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The new shifting-plus-antilexicographic-compression transformation preserves both the minimum-degree condition and the extremal property for arbitrary k > ℓ ≥ 2.","pith_extraction_headline":"For vertex sets larger than a quadratic bound in t, any extremal k-uniform hypergraph with minimum degree binom(t, k-1) contains an isolated copy of K^k_{t+1}."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.02610/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T15:37:42.838309Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T03:01:22.177127Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T16:11:33.542519Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"98d39a171c08337f7f0838f3290d9b29a74911fb12f1a6f210a5056c36ce0cd0"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"bf89b30b9cd7ad68fc0b65454a3913436ca377fade533e620370ed332a0a76b2"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}