# Cardinals realizable by the chromatic number of a regular hypergraph

For any set $$X$$ and cardinal $$\kappa$$, we denote by $$[X]^\kappa$$ the collection of subsets of $$X$$ having cardinality $$\kappa$$.

If $$H=(V,E)$$ is a hypergraph, and $$\kappa$$ is a cardinal, we say that a map $$c:V\to \kappa$$ is a coloring if for every $$e\in E$$ with $$|e|\geq 2$$ the restriction $$c|_e$$ is non-constant. By $$\chi(H)$$ we denote the minimum cardinal such that there is a coloring $$c:V\to \kappa$$.

Let cardinals $$\kappa\geq \omega$$ and $$\alpha,\beta$$ with $$2< \alpha, \beta \leq \kappa$$ be given.

Is there necessarily $$E \subseteq [\kappa]^\beta$$ such that $$\chi(\kappa,E) = \alpha$$? (There is an easy positive answer for either $$\alpha$$ or $$\beta$$ equal to $$2$$.)

• Could you confirm that you mean non-constant rather than injective? For example, if I have a 3-edge with {a,b,c}, then I can color both a and c red, and b blue? – Joel David Hamkins Jan 3 at 15:44
• @JoelDavidHamkins it is standard terminology in the hypergraph colorings theory – Fedor Petrov Jan 4 at 15:54
• @FedorPetrov Thanks very much. – Joel David Hamkins Jan 4 at 17:01
• "Injective on edges" is the wrong generalisation of proper colouring to hypergraphs because it reduces to graph colouring by replacing hyperedges by graph cliques, whereas "non-constant" is more expressive. – Ben Barber Jan 5 at 15:15
• For $\kappa=\beta=\aleph_\omega$, no subgraph of $[\kappa]^\beta$ has chromatic number greater than $\omega$, since we can color $\gamma\in\kappa$ with $n$ if $\aleph_{n-1}\leq\gamma<\alpha_n$, taking $\alpha_{-1}=0$ for convenience. I suspect the answer might be positive whenever $\alpha\leq cf(\kappa)$. – Wojowu Jan 6 at 15:34