# Size of edge set of infinite hypergraphs with $\chi(H) = |V(G)|$

Let $$H=(V,E)$$ be a hypergraph such that for every $$e\in E$$ we have $$|e|\geq 2$$. A map $$c:V\to \kappa$$, where $$\kappa$$ is a cardinal, is said to be a (hypergraph) coloring if for all $$e\in E$$ the restriction $$c|_e$$ is not constant. By $$\chi(H)$$ we denote the smallest cardinal $$\kappa$$ such that there is a coloring map $$c:V\to \kappa$$.

Is there a hypergraph $$H=(V,E)$$ with $$V$$ infinite, $$|E|<|V|$$ and $$\chi(H) = |V|$$?

EDIT: My original answer (below) is correct but not really optimal. This is easier: simply well-order the set of vertices and greedily color them using the well-ordering. More precisely, if you reach a vertex $$v$$ and have already colored all vertices appearing before $$v$$ in the well-ordering, color $$v$$ with the smallest ordinal that does not create a monochromatic edge with what has been done before. Since $$v$$ is contained in at most $$|E|$$ edges, there are at most $$|E|$$ forbidden colors, so the range of this coloring is at most $$|E| + 1$$. Thus, $$\chi(H) \leq |E| + 1$$ (or, sometimes better, the maximum degree of the hypergraph plus 1).
Original answer: No, this is impossible. It is always the case that $$\chi(H) \leq |E|$$ (or at least $$\chi(H) \leq 2|E|$$ if $$E$$ is finite). To see this, note that it's easy to define an injective partial function $$c_0$$ from a subset of $$V$$ to $$|E|$$ (or $$2|E|$$ if $$E$$ is finite) such that, for all $$e \in E$$, we have $$|\mathrm{dom}(c_0) \cap e| \geq 2$$ (simply enumerate the edges in order type $$|E|$$ and take care of them one at a time). Then any extension of $$c_0$$ to a function $$c:V \rightarrow |E|$$ will be a hypergraph coloring of $$H$$.