Worked solution: Kang–Kelly–Kühn–Methuku–Osthus asymptotic proof via iterative absorption (2021)

Kang, Kelly, Kuhn, Methuku and Osthus (2023, Introduction) recall that in 1972 Erdős, Faber and Lovász conjectured three equivalent statements (see Section 1.1), of which the graph-theoretic reading is: if a graph $G$ is the union of $n$ cliques, each of size at most $n$, such that every pair of cliques shares at most one vertex, then the chromatic number of $G$ is at most $n$. The equivalent hypergraph-colouring formulation, which the paper works with throughout, is: for a linear hypergraph $\mathcal{H}$ (one where any two edges share at most one vertex) on $n$ vertices, the chromatic index $\chi'(\mathcal{H})$ -- the minimum number of colours needed to colour the hyperedges so that intersecting edges get different colours -- satisfies $\chi'(\mathcal{H}) \le n$. Erdős called this one of his three favourite combinatorial problems and, as its difficulty became apparent, offered escalating rewards that eventually reached 500 dollars.