Colorings of uniform hypergraphs with large girth

Abstract

This work deals with a wellknown extremal com� binatorial problem concerning colorings of uniform hypergraphs with large girth. First, we recall the basic concepts from hypergraph theory. A hypergraph is a pair of sets H =( V, E), where V is a finite set known as the vertex set of the hypergraph and E is a collection of subsets of V, with these subsets called the edges of the hypergraph. A hypergraph is nuniform if each of its edges contains exactly n vertices. A coloring of the vertex set V in H = (V, E) is said to be proper if all the edges in E are not monochro� matic. The chromatic number of H is the minimum number of colors required for a proper coloring of its vertex set. The chromatic number of H is denoted by χ(H). A cycle of length k in H = (V, E) is an alternating sequence v 0 , e 1 , v 1 , …, e k , v k = v 0 consisting of k dif� ferent vertices v0, v1 ,… , vk -1 and k different edges e1, e2, …, ek such that vi -1 ∈ ei and vi ∈ ei for any i = 1, 2, …, k. The length of the minimum cycle in H is called its girth and is denoted by g(H). The degree of a vertex v in H is the number of edges in H that contain v. The maximum vertex degree in a hypergraph is denoted by Δ(H). The study of the interrelation between the chro� matic number, girth, and maximum vertex degree in nuniform hypergraphs was begun in Erdos and Lovasz's classical work (1). They showed that, if an n� uniform hypergraph has a large chromatic number, then its maximum vertex degree cannot be very low. Thereby, they motivated the study of the extremal value Δ(n, r, s) equal to the minimum possible value of the maximum vertex degree of am nuniform hyper� graph with a chromatic number greater than r and girth greater than s. Formally, Δ(n, r, s) is defined as This definition implies that