Claude Tardif: The chromatic number of the product of 15-chromatic graphs can be 14

Abstract: Hedetniemi's conjecture stated (or states, if a conjecture is intemporal) that the chromatic number of a product of graphs is the minimum of the chromatic number of the factors. Last year, Yaroslav Shitov demolished this 53 year old conjecture in a 3 page paper. The chromatic numbers involved in his counterexamples were very large. However a few months ago found a way to avoid the asymptotic arguments used by Shitov, using instead some injective colourings of graphs at a crucial point of the argument. This allowed Xuding Zhu to show that the chromatic numbers of the counterexamples can be reasonably small. In his paper, the bound for the factors was 225, which may be further reduced if graphs exist with odd girth 7 and a large enough fractional chromatic number. Here we modify the argument again. Instead of injective colourings, we use colourings that are "wide'' in the sense of Simonyi and Tardos. In this way, we can prove the result in the title. Depending on the chromatic number of a certain graph with 13965 vertices, it is possible that the same argument gives a product of 12-chromatic graphs with chromatic number 11.