Solution to a problem of Erdős on the chromatic index of hypergraphs with bounded codegree

In 1977, Erdős asked the following question: for any integers (Formula presented.), if (Formula presented.) are complete graphs such that each (Formula presented.) has at most (Formula presented.) vertices and every pair of them shares at most (Formula presented.) vertices, what is the largest possible chromatic number of the union (Formula presented.) ? The equivalent dual formulation of this question asks for the largest chromatic index of an (Formula presented.) -vertex hypergraph with maximum degree at most (Formula presented.) and maximum codegree at most (Formula presented.). For the case (Formula presented.), Erdős, Faber and Lovász famously conjectured that the answer is (Formula presented.), which was recently proved by the authors for all sufficiently large (Formula presented.). In this paper, we answer this question of Erdős for (Formula presented.) in a strong sense, by proving that every (Formula presented.) -vertex hypergraph with maximum degree at most (Formula presented.) and maximum codegree at most (Formula presented.) has chromatic index at most (Formula presented.) for any (Formula presented.). Moreover, equality holds if and only if the hypergraph is a (Formula presented.) -fold projective plane of order (Formula presented.), where (Formula presented.). Thus, for every (Formula presented.), this bound is best possible for infinitely many integers (Formula presented.). This result also holds for the list chromatic index.