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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 $V=\binom\omega{k-1}$ and $E=\{\binom X{k-1}:X\in\binom\omega k\}$.

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 $\chi(H_n)\gt n$.

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 $\chi(H_n)\gt n$.

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 $V=\binom\omega{k-1}$ and $E=\{\binom X{k-1}:X\in\binom\omega k\}$.

It follows from Ramsey's theorem that $\chi(H)\gt n$ for each $n\lt\omega$; the other properties are obvious.

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 $V=\binom\omega{k-1}$ and $E=\{\binom X{k-1}:X\in\binom\omega k\}$.

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 $\chi(H_n)\gt n$.

That is false 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 $V=\binom\omega{k-1}$ and $E=\{\binom X{k-1}:X\in\binom\omega k\}$.

It follows from Ramsey's theorem that $\chi(H)\gt n$ for each $n\lt\omega$; the other properties are obvious.

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 $V=\binom\omega{k-1}$ and $E=\{\binom X{k-1}:X\in\binom\omega k\}$.

It follows from Ramsey's theorem that $\chi(H)\gt n$ for each $n\lt\omega$; the other properties are obvious.