Solution to a problem of Erd\H{o}s on the chromatic index of hypergraphs with bounded codegree

TL;DR

This paper resolves Erdős's longstanding question on the chromatic index of hypergraphs with bounded codegree, establishing tight bounds and characterizing extremal cases for large hypergraphs.

Contribution

It proves that hypergraphs with maximum degree near $tn$ and codegree at most $t$ have chromatic index at most $tn$, confirming a conjecture for all large $n$ and characterizing extremal structures.

Findings

Chromatic index at most $tn$ for hypergraphs with specified degree and codegree bounds.

Equality holds only for hypergraphs that are $t$-fold projective planes.

The bounds are tight for infinitely many values of $n$.

Abstract

In 1977, Erd\H{o}s 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 $t = 1$, Erd\H{o}s, Faber, and Lov\'{a}sz 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 Erd\H{o}s 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.