Coloring linear hypergraphs: the Erdos-Faber-Lovasz conjecture and the Combinatorial Nullstellensatz

The long-standing Erdos-Faber-Lovasz conjecture states that every n-uniform linear hypergaph with n edges has a proper vertex-coloring using n colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdos-Faber-Lovasz conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work

A proof of the Erd\H{o}s-Faber-Lov\'asz conjecture

The Erd\H{o}s-Faber-Lov\'{a}sz conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stability versions of this result, which confirm a prediction of Kahn.Comment: 39 pages, 2 figures; this version includes additional references and makes two small corrections (definition of a useful pair in Section 5 and an additional condition in the statement of Lemma 6.2