Worked solution: Kang–Kelly–Kühn–Methuku–Osthus asymptotic proof via iterative absorption (2021)
Every piece of the puzzle is now in place: whether the linear hypergraph resembles a finite projective plane, a near-complete graph , a near-pencil, or sits far away from all three extremal shapes, the stratified colouring pipeline packs all its hyperedges into at most colour classes.
What Erdős, Faber and Lovász posed at a party in 1972 as a seemingly modest test problem for hypergraph theory is thus resolved for all sufficiently large by uniting fifty years of probabilistic combinatorics -- the Rödl nibble, locally sparse graph colouring, vertex absorption, and -factorisation.
Kang, Kelly, Kühn, Methuku and Osthus (2023, Section 8) combine the case divisions of Sections 5, 6, and 7 to complete the proof of Theorem 1.1: there exists such that for every , every linear hypergraph on vertices has chromatic index . By duality, this simultaneously proves all three equivalent formulations from Section 1.1: edge-colouring linear hypergraphs, vertex-colouring unions of cliques of size at most sharing at most one vertex pairwise, and rainbow-type set colourings of sets of size with pairwise intersection at most .
As the authors note in Section 1.2, every probabilistic step of the proof can also be derandomised into a randomized polynomial-time colouring algorithm for large .