We say that a hypergraph $H=(V,E)$ has *property ${\bf B}$* if there is $S\subseteq V$ such that for all $e\in E$ with $|e|\geq 2$ we have $$(e\cap S) \neq \emptyset \neq (e\cap (V\setminus S)).$$

If $k\in \omega$, a hypergraph $H=(V,E)$ is $k$-*uniform* if all elements of $E$ have cardinaliy $k$.

Suppose the hypergraph $(\omega,E)$ is $k$-uniform for some $k\geq 4$, and we have that $$2\cdot |e_0\cap e_1| < k$$ whenever $e_0\neq e_1\in E$. Does this necessarily imply that $(\omega,E)$ has property ${\bf B}$?