May the same graph admit two different chromatic numbers in two different universes? How about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Gödel’s constructible universe, for every uncountable cardinal $$\mu $$ below the first fixed-point of the $$\aleph $$ -function, there exists a graph $$\mathcal G_\mu $$ satisfying the following.