Summary: Addendum to:
Aaron Archer ``On the upper chromatic numbers of the reals,''
Discrete Math., 214 (2000) 6575.
In the journal paper, I included the following theorem without proof, since the proof is very similar to
Erdos's probabilistic proof that there exist graphs with arbitrarily high girth and chromatic number (as it
appears in [1, p. 35]). Since some people have inquired about the proof, I set it down here for completeness.
Theorem 14. Given any l, m # N there exists a graph G with girth(G) > l and # cap (G) # m.
Proof: Given a graph G = (V, E) with its edges colored from the set [m], let E i denote the set of edges
that are colored i, and let G i = (V, E i ), i.e., the subgraph of G induced by the edges colored i. Let #(G i )
denote the independence number of G i , in other words, the maximum number of vertices of G that can be
colored i in a vertex coloring compatible with G's edge coloring.
We will show using the probabilistic method that there exists a simple edgecolored graph G = (V, E)
with a small number of cycles of length at most l, such that #(G i ) is small for each i. We then delete an
arbitrary edge from each such short cycle, yielding a new graph G # . By construction, girth(G # ) > l. Let D
denote the set of deleted edges. Since #(G # i ) # #(G i )+|D#E i |, # i #(G # i ) is still small. Choosing appropriate
parameters of our probabilistic construction, we will get # i #(G # i ) < n (where n = |V |), which implies that
G # has no compatible vertex coloring.
Let # < 1
l , p = mn #-1 . Let G be a random graph on n vertices, where each of the # n
2 # possible edges