{ "artifactId": "artifact.library.mathlib.erdos-ko-rado-theorem.v001", "candidateOnly": false, "category": "Combinatorics", "claimBoundary": "This page indexes Mathlib’s finite-family theorem for 𝒜 : Finset (Finset (Fin n)). The family is pairwise intersecting, every member has exactly r elements, and the hypothesis is exactly r ≤ n / 2. The selected declaration proves only #𝒜 ≤ (n−1).choose (r−1); it does not construct a star family, prove a separate sharpness result, or classify equality cases.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem Finset.erdos_ko_rado {n : ℕ} {𝒜 : Finset (Finset (Fin n))} {r : ℕ} (h𝒜 : (𝒜 : Set (Finset (Fin n))).Intersecting) (h₂ : (𝒜 : Set (Finset (Fin n))).Sized r) (h₃ : r ≤ n / 2) : #𝒜 ≤ (n - 1).choose (r - 1)", "family": "Extremal set theory", "id": "library.mathlib.erdos-ko-rado-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/erdos-ko-rado-theorem/", "entryData": "/data/library-theorems/erdos-ko-rado-theorem.json", "evidence": "/proofs/artifact.library.mathlib.erdos-ko-rado-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.erdos-ko-rado-theorem.v001.evidence.json", "source": "/sources/upstream/erdos-ko-rado-theorem/" }, "landmark": { "completeFormalType": "theorem Finset.erdos_ko_rado {n : ℕ} {𝒜 : Finset (Finset (Fin n))} {r : ℕ} (h𝒜 : (𝒜 : Set (Finset (Fin n))).Intersecting) (h₂ : (𝒜 : Set (Finset (Fin n))).Sized r) (h₃ : r ≤ n / 2) : #𝒜 ≤ (n - 1).choose (r - 1)", "concepts": [ "intersecting set families", "uniform set families", "extremal combinatorics", "iterated shadows", "Kruskal–Katona theorem" ], "plainLanguage": "Choose many r-element subsets of an n-element ground set, requiring every two chosen subsets to share at least one element. When r is no more than half of n, Erdős–Ko–Rado gives the numerical ceiling binom(n−1,r−1) for the number of chosen subsets.", "poster": { "alt": "An ivory extremal-set-theory poster states the pairwise-intersecting uniform-family bound and traces complements, an iterated shadow, disjointness, Kruskal–Katona, and Pascal counting.", "byteSize": 2132146, "caption": "Complements and an iterated-shadow bound turn pairwise intersection into the cardinality obstruction behind Erdős–Ko–Rado.", "description": "The poster makes the three hypotheses and upper bound primary, using a three-card example whose pairwise overlaps are different. Its source route takes complements, descends by an iterated shadow, uses disjointness from the original family, invokes the Lovász Kruskal–Katona bound, and finishes with Pascal’s binomial identity. The footer excludes equality classification.", "derivatives": [ { "byteSize": 81594, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/erdos-ko-rado-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:bb2bb389cc239906c73f801108c0aa15c413841fa96336a4afcb3361d6f72daf", "width": 640 }, { "byteSize": 143872, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/erdos-ko-rado-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:9d3398187e7b5b1cc24d76685f3c221d978a214a1d00c0cb70a738e8c09e7963", "width": 1024 } ], "generatedAt": "2026-07-25T16:18:13.288520905-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/erdos-ko-rado-theorem/theorem-poster-v1.png", "sha256": "sha256:a9a7978451f79b6025448c6d2379f4bc90a4383680ec329b9125df762d3c3e70", "title": "Erdős–Ko–Rado at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "EXTREMAL SET THEORY\nERDŐS–KO–RADO\nEVERY TWO MEMBERS MEET\nEVERY MEMBER HAS SIZE r\nr ≤ n/2\n|A| ≤ binom(n−1, r−1)\nCOMPLEMENTS\nITERATED SHADOW\nDISJOINTNESS\nKRUSKAL–KATONA\nPASCAL COUNT\nUPPER BOUND ONLY\nNo equality classification here." }, "visual": { "alt": "Three uniform set cards each contain three tokens; each pair shares a different token, no token lies in all three cards, and the Erdős–Ko–Rado upper bound appears below.", "byteSize": 2356946, "caption": "Pairwise intersection plus uniform size r and r ≤ n/2 forces |𝒜| ≤ binom(n−1,r−1).", "description": "Three cards encode the illustrative family {circle, square, triangle}, {triangle, diamond, hexagon}, and {circle, hexagon, star}. Gold connectors identify the three different pairwise overlaps, making clear that pairwise intersection does not require one element common to the whole family.", "derivatives": [ { "byteSize": 27038, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/erdos-ko-rado-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:7fbdbf24ad93c015d2c420830cfcfe3f00d52da1dec01d171597ead328ac138e", "width": 640 }, { "byteSize": 61554, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/erdos-ko-rado-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:d1984c8a29ae553cce5c36eaaced4ed9b03ddc3e72e701ba596539be3c17ad90", "width": 1200 } ], "generatedAt": "2026-07-25T16:18:13.288520905-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/erdos-ko-rado-theorem/theorem-schematic-v1.png", "sha256": "sha256:618fb644e8f1de36dcb76f68a3025ffd96ad69b74a4764537f040bcb1dca5b86", "title": "Pairwise-intersecting uniform-family schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "Erdős–Ko–Rado is a foundational extremal set theorem. Mathlib derives its upper bound through complements, an iterated shadow, disjointness, and the Lovász form of Kruskal–Katona, giving the page a theorem-specific checked proof architecture." }, "nonClaims": [ "ProofAtlas did not originate the Erdős–Ko–Rado theorem or Mathlib’s declaration.", "The selected declaration is not a theorem about arbitrary infinite set families.", "It does not assert that all members share one common element.", "It does not formalize an extremal construction or classify equality cases.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.erdos-ko-rado-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "A pairwise-intersecting family of r-element subsets of an n-element set, with r at most half of n, has at most binom(n−1,r−1) members.", "targetId": "target.library.mathlib.erdos-ko-rado-theorem.v001", "title": "Erdős–Ko–Rado Theorem", "upstreamOrigin": { "declarationName": "Finset.erdos_ko_rado", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Combinatorics/SetFamily/KruskalKatona.lean#L347", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:c6351d7ee422db9eed8f45335f4128eb3a66fe09997d12abc15eba38e9863f1c", "sourceByteLength": 19772, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Combinatorics/SetFamily/KruskalKatona.lean", "sourceLine": 347, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }