In 2019, Shitov has shown a counterexample ([Ann. Math, **190**(2) (2019) pp. 663-667](https://doi.org/10.4007/annals.2019.190.2.6)) to [Hedetniemi’s conjecture](https://en.wikipedia.org/wiki/Hedetniemi%27s_conjecture),


$$\chi(G \times H)=\min(\chi(G),\chi(H))$$
where $\chi(G)$ is the [chromatic number](https://en.wikipedia.org/wiki/Chromatic_number) of the undirected finite graph $G$.


Shitov counterexample is estimated to have $|V(G)|\approx4^{100}$ and $|V(H)|\approx4^{10000}$. Has there been some effort or progress to reduce the size of the counterexample?



Smallest known counterexamples to Hedetniemi’s conjecture
Yes, Xuding Zhu did this in *Relatively small counterexamples to Hedetniemi's conjecture* (J. Comb. Theory B **146** (2021) pp. 141-150, doi:[10.1016/j.jctb.2020.09.005](https://doi.org/10.1016/j.jctb.2020.09.005), arXiv:[2004.09028](https://arxiv.org/abs/2004.09028)) where the sizes of the graphs are $3403$ and $10501$.


Marcin Wrochna has a preprint, *Smaller counterexamples to Hedetniemi's conjecture*, arXiv:[2012.13558](https://arxiv.org/abs/2012.13558), that brings the sizes down to $4686$ and $30$ (as well as the chromatic number down to $5$).


