Erdős–Faber–Lovász conjecture

[1] It says: The conjecture for all sufficiently large values of k was proved by Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus.

The n cliques of size n in the Erdős–Faber–Lovász conjecture may be interpreted as the hyperedges of an n-uniform linear hypergraph that has the same vertices as the underlying graph.

In the graph coloring formulation of the Erdős–Faber–Lovász conjecture, it is safe to remove vertices that belong to a single clique, as their coloring presents no difficulty; once this is done, the hypergraph that has a vertex for each clique, and a hyperedge for each graph vertex, forms a simple hypergraph.

Paul Erdős, Vance Faber, and László Lovász formulated the harmless looking conjecture at a party in Boulder Colorado in September 1972.

[1] Paul Erdős originally offered US$50 for proving the conjecture in the affirmative, and later raised the reward to US$500.

In 2023, almost 50 years after the original conjecture was stated,[1] it was resolved for all sufficiently large n by Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus.

He shows that, for any fixed value of L, a finite calculation suffices to verify that the conjecture is true for all simple hypergraphs with that value of L. Based on this idea, he shows that the conjecture is indeed true for all simple hypergraphs with L ≤ 10.