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

Abstract

In 1977, Erdős asked the following question: for any integers $t,n \in \mathbb{N}$, if $G_1 , \dots , G_n$ are complete graphs such that each $G_i$ has at most $n$ vertices and every pair of them shares at most $t$ vertices, what is the largest possible chromatic number of the union $\bigcup_{i=1}^{n} G_i$? The equivalent dual formulation of this question asks for the largest chromatic index of an $n$-vertex hypergraph with maximum degree at most $n$ and maximum codegree at most $t$. For the case $t = 1$, Erdős, Faber, and Lovász famously conjectured that the answer is $n$, which was recently proved by the authors for all sufficiently large $n$. In this paper, we answer this question of Erdős for $t \geq 2$ in a strong sense, by proving that every $n$-vertex hypergraph with maximum degree at most $(1-o(1))tn$ and maximum codegree at most $t$ has chromatic index at most $tn$ for any $t,n \in \mathbb{N}$. Moreover, equality holds if and only if the hypergraph is a $t$-fold projective plane of order $k$, where $n = k^2 + k + 1$. Thus, for every $t \in \mathbb N$, this bound is best possible for infinitely many integers $n$. This result also holds for the list chromatic index.

Theorems & Definitions (54)

• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 3.1: Kahn kahn1996asymptotically
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• proof