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{
"table": "T423",
"baseline_revision": "3884a168c9d18a82f5d1d7be1a7fb0c2455aa2bb35c1d597bca9ad7dd7369da5",
"baseline_entries": 67,
"status": "arithmetic hypergraph edge-size family and branch endpoint",
"selection": "it holds the branch endpoint $u=1$ and the positive roots at $u=k-1$ for hypergraph edge sizes $3\\leq k\\leq13$; these integer arguments are selected by the $2$-core minimisation problem CITE{CoreMinimisation}",
"reason": "For a $k$-uniform random hypergraph, the $2$-core threshold minimises a constant multiple of $\\lambda/(1-e^{-\\lambda})^{k-1}$ over $\\lambda>0$ CITE{CoreMinimisation}, equation (1) with $r=2$. Differentiation gives $e^\\lambda-1=(k-1)\\lambda$, selecting $u=k-1$. This derivation, not the fact that the arguments are integers, explains the finite prefix $3\\leq k\\leq13$. The endpoint $u=1$ remains exact; zero is excluded from the positive-root branch when $u>1$.",
"primary_sources": [
"https://www.cs.toronto.edu/~molloy/webpapers/xorrsarevised.pdf",
"https://numdam.org/item/JTNB_1993__5_2_411_0.pdf"
],
"skill_sha256": "b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18",
"earlier_skill_sha256": "ddda1bedfde5a883d9a3d1d24e88e731970776abd37e4a91d4c01c4af8542c44",
"every_free_parameter": {
"u": "arithmetic hypergraph edge-size family and branch endpoint"
},
"retained_numeric_identifiers_reused_without_reassignment": true,
"hand_edited_leaf_metadata_preserved": true,
"removed_samples_retained_in_website_revision_history": true,
"publication_or_approval_requested": false
}