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

Kang, Kelly, Kühn, Methuku and Osthus (2023, Section 1.1) begin from three formulations that Erdős, Faber and Lovász conjectured in 1972. The one the paper works with throughout is the hypergraph form: every linear hypergraph $\mathcal{H}$ on $n$ vertices — meaning $|e \cap f| \le 1$ for every pair of distinct edges — has chromatic index $\chi'(\mathcal{H}) \le n$, where $\chi'(\mathcal{H})$ counts the colours needed on edges so that intersecting edges differ. The second form is vertex-colouring: if a graph $G$ is the union of $n$ cliques, each of size at most $n$ and pairwise sharing at most one vertex, then $\chi(G) \le n$. The third is the original set version: given $n$ sets of size $n$ with pairwise intersections of size at most $1$, their elements can be coloured with $n$ colours so that every set is rainbow.

Equivalence is pure duality. Proper edge-colourings of $\mathcal{H}$ are proper vertex-colourings of its line graph $L(\mathcal{H})$, in which each vertex $v$ of $\mathcal{H}$ induces a clique on the edges containing $v$, and linearity is exactly the 'share at most one vertex' hypothesis. The transpose of the set incidence matrix interchanges the set and clique readings. All quantitative reasoning in the proof therefore takes place on $\mathcal{H}$, whose degree and codegree data drive the nibble machinery.

The bound $n$ is forced, and by three genuinely different witnesses recalled in Section 1.1: the complete graph $K_n$ with $n$ odd, whose line graph is $n$-chromatic; the near-pencil, one edge on $n-1$ vertices plus $n-1$ edges through its missing vertex; and a finite projective plane, about $n$ mutually intersecting edges of size about $\sqrt{n}$. These extremal families reappear throughout as the cases with no slack.