# Uniform hypergraphs with small edge intersections and propery ${\bf B}$

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 $${\bf B}$$?

• "$k$-uniform", not "$k$-regular". $k$-regularity is the dual property that every vertex lies in $k$ edges. Jan 27, 2023 at 11:06
• Thanks - will correct Jan 27, 2023 at 13:22

There are counterexamples for every integer $$k\ge3$$.
In fact, if $$2\le k\lt\omega$$, there is a $$k$$-uniform hypergraph $$H=(V,E)$$ such that $$|V|=\aleph_0$$, $$\{e_1,e_2\}\in\binom E2\implies|e_1\cap e_2|\le1$$, and $$H$$ has chromatic number $$\chi(H)=\aleph_0$$.
Namely, let $$V=\binom{\mathbb N}{k-1}$$ and $$E=\{\binom X{k-1}:X\in\binom{\mathbb N}k\}$$.
It follows from Ramsey's theorem that $$\chi(H)\gt n$$ for each $$n\lt\omega$$; the other properties are obvious. Moreover, $$H$$ has a finite subhypergraph $$H_n$$ with $$\chi(H_n)\gt n$$.