Coloring hypergraphs of small codegree, and a proof of the Erdős–Faber–Lovász conjecture.

The theory of edge-coloring hypergraphs has a rich history with important connections and application to other areas of combinatorics e.g. design theory and combinatorial geometry.  A long-standing problem in the field is the Erdős–Faber–Lovász conjecture (posed in 1972), which states that the chromatic index of any linear hypergraph on n vertices is at most n.  In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large n.  Recently, we also solved a related problem of Erdős from 1977 on the chromatic index of hypergraphs of small codegree.  In this talk, I will survey the history behind these results and discuss some aspects of the proofs.  I will also discuss some algorithmic aspects of these results.