Counterexamples to Hedetniemi?s conjecture and infinite Boolean lattices

We prove that for any c >= 5, there exists an infinite family (G(n))(n is an element of N )of graphs such that chi(G(n)) > c for all n E N and chi(G(m) x G(n)) <= c for all m &NOTEQUexpressionL; n. These counterex-amples to Hedetniemi's conjecture show that the Boolean lattice of exponential graphs with K-c as a base is infinite for c >= 5.