A Construction of Graphs without Triangles having Pre‐Assigned Order and Chromatic Number

The chromatic number x(r) of a combinatorial graph r is the least cardinal number a such that the set of nodes of I’ can be divided into a subsets so that every edge of P joins nodes belonging to different subsets. It is known? that corresponding to every finite a there exists a finite graph Pa without triangles satisfying x(P,) = a. In [l], Theorem 2, we have extended this result to transflnite values of a. For every graph F the order 4 (l?), i.e. the cardinal of the set of nodes of I’, satisfies 4 (I’) > x( I’). The construction used in [l] was of considerable complexity and did not allow us to prove that it was most economical, i.e. that it leads to a graph Pa such that +(I’,) = a. This equation was only established ([l], Theorem 3) when essential use was made of a form of the general continuum hypothesis. In the present note we describe a much simpler construction of such a graph Pa and we shall at the same time prove, without using the continuum hypothesis, that our new graph Pa satisfies $(l’,) = x(l’J = a. Trivially, for instance by adding isolated nodes to the graph, we can make its order equal to any given cardinal b such that b 3 a, without changing the chromatic number or introducing any triangles.