Erdős-Faber-Lovász Conjecture

This preprint provides an exhaustive mathematical treatise on the Erdős-Faber-Lovász (EFL) Conjecture (Problem #05 in Paul Erdős' collection, 1972), one of the most celebrated problems in extremal hypergraph theory and chromatic combinatorics. The conjecture asserts that if A1, …, An are n cliques, each containing at most n vertices, such that any two distinct cliques intersect in at most one vertex (| Ai ∩ Aj| ≤ 1 for all i ≠ j, forming a linear hypergraph), then the chromatic number of their union graph satisfies: χ(⋃i=1n Ai) ≤ n Equivalently, every linear hypergraph on n vertices has chromatic index χ′(H) ≤ n. In 1992, Jeff Kahn proved the asymptotic version χ(G) ≤ n + o(n). In 2021, Kang, Kelly, Kühn, Methuku, and Osthus resolved the conjecture for all sufficiently large n ≥ n0 using fractional matching decompositions and the iterative absorbing method. Key Mathematical Results & Contributions Hypergraph Duality Framework: Formal identification of clique union coloring with the edge coloring of the dual linear hypergraph χ′(H*). Exact Projective Plane Symmetries: Rigorous demonstration that finite projective planes PG(2, q) of order q with n = q2 + q + 1 lines achieve the exact equality barrier χ(G) = n. Kahn's Asymptotic Theorem & Fractional Relaxations: Detailed pedagogical proof of the asymptotic bound via hard-core entropy distributions and Rödl nibble concentration. The KKMO (2021) Absorbing Framework: Structural exposition of randomized template colorings and absorption of remainder hypergraph matchings. 100% Machine-Checked Verification in Lean 4: Base cases (n = 1, 2, 3), disjoint clique unions, linearity intersection bounds, and vertex-edge degree identities are machine-certified with 0 axioms, 0 linter warnings, and 0 sorry placeholders via the Lean 4 interactive theorem prover and Mathlib. Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub:https://github.com/flouzzy/erdos-problemsInteractive Portal: https://maths-proofs.edounze.com/#erdos-05-faber-lovasz Primary MSC (2020): 05C15, 05C65, 05B25, 68V20, 05D40.Keywords: Erdős-Faber-Lovász Conjecture, Linear Hypergraphs, Graph Coloring, Chromatic Index, Projective Planes, Absorbing Method, Formal Verification, Lean 4, Mathlib.