Erdős–Faber–Lovász conjecture

If $G = \bigcup_{i=1}^n A_i$ is the union of $n$ edge-disjoint complete graphs $A_1, \dots, A_n$, each on $n$ vertices, such that $|V(A_i) \cap V(A_j)| \le 1$ for all $1 \le i < j \le n$, then the chromatic number of $G$ satisfies $\chi(G) = n$ (equivalently, every linear hypergraph on $n$ vertices has chromatic index $\chi'(\mathcal{H}) \le n$).

Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus announced a proof in January 2021 (published in the Annals of Mathematics in 2023) establishing the Erdős–Faber–Lovász conjecture for all sufficiently large $n \ge n_0$. Strictly speaking, the problem across all integers $n$ is solved asymptotically (for $n \ge n_0$) and verified for $n \le 12$ (Hindman 1981 for $n \le 10$; Romero and Alonso-Pecina 2014 for $n \le 12$), while finite intermediate values $13 \le n < n_0$ remain open.