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

In a general linear hypergraph $\mathcal{H}$, hyperedges can have any size from $2$ up to $n$: a projective plane has edges of size around $\sqrt{n}$, while a near-pencil has one giant edge of size $n-1$. No single colouring tool works across all scales, because two giant edges almost always collide at a shared vertex, whereas tiny edges rarely collide.

The authors therefore slice $\mathcal{H}$ into size regimes -- very large edges, edges near the projective-plane scale $\sqrt{n}$, medium edges, and bounded-size small edges -- and colour them from largest down to smallest so that the hardest collisions are resolved first while all $n$ colours are still available.