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On Chromatic Number of Graphs and Set-systems

P. ERDŐS, L. KALMÁR

Abstract

Let a be a cardinal number . A graph V is said to have chromatic number a if x is the least cardinal such that, the set of vertices of ~ is the union of a sets, where no two elements of the same set are connected by an edge in ~§ . A graph 16~ is said to have colouring-number a, if a is the least cardinal such that the set of vertices of S has a well-ordering -< satisfying the condition that for every vertex x of W the number of those vertices y <x of .§ which are adjacent to x is less than a . We consider a graph ~rt as an ordered couple (g, G where g is a set the elements of which are called the vertices of W, and G is a subset of the set of all unordered pairs of g. The elements of G are called the edges of S . As a generalization of graphs we will consider set-systems if =(h, H) where h is a set and H is a set of subsets ofh . It is easy to generalize the concepts of chromatic and colouring numbers for general set systems instead of graphs . We mention that several properties of set-systems have been investigated in the literature which can be expressed using the notion of chromatic-number . We do not try to give complete references but we point out one important property which can be expressed using it . In [11] MILLER defined property B of a set of sets . A set H of sets is said to have property B if there exists a set B which meets each element of H but does not contain any of them . In [5] we have investigated property B in greater detail . It is obvious that a non-empty set of sets H has property B iff the corresponding set-system _ ( U H, H) has chromatic number 2 . Our main aim in this paper is to study the colouringand chromatic-numbers of (infinite) graphs . The introduction of set-systems has several purposes . First we are going to generalize some easy theorems proved in this paper for general set-systems . Meantime we will state and prove some theorems for set-systems which will serve as lemmas to prove the results for graphs . Finally we will give some generalizations of known theorems concerning finite graphs, for more general finite set-sytems .

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