On the chromatic number and independence number of hypergraph products

The hypergraph product G□H has vertex set V(G)×V(H), and edge set , where × denotes the usual Cartesian product of sets. We construct a hypergraph sequence {Gn} for with χ(Gn)→∞ and χ(Gn□Gn)=2 for all n. This disproves a conjecture of Berge and Simonovits [C. Berge, M. Simonovits, The coloring numbers of the direct product of two hypergraphs, in: Hypergraph Seminar, Proc. First Working Sem., Ohio State Univ., Columbus, OH, 1972; Dedicated to Arnold Ross, in: Lecture Notes in Math., vol. 411, Springer, Berlin, 1974, pp. 21–33]. On the other hand, we show that if G and H are hypergraphs with infinite chromatic number, then the chromatic number of G□H is also infinite.We also provide a counterexample to a “dual” version of their conjecture, by constructing a graph sequence {Gn} with α(Gn)/|V(Gn)|→0 and α(Gn□Gn)/2|V(Gn)|→1/2. The constant 1/2 cannot be replaced by a larger number.