Hypergraph with the property that for any two edges $F_1, F_2 \in \mathcal F, |F_1 ∩ F_2 | ≥ 2$ is 2-colourable

Question

Let $\mathcal F$ be a hypergraph with the property that for any two edges $F_1, F_2 \in \mathcal F, |F_1 ∩ F_2 | ≥ 2$. Prove that $\mathcal F$ is two-colourable.

I have no idea to prove the claim. Can anyone give me some hints?

Answer 1

This is problem 33 in chapter "§13 Hypergraphs" in the book "Combinatorial Problems and Exercises" by László Lovász:

We construct the 2-colouring (black, white) of the vertex set $V=\{v_1,...,v_n\}$ step by step. Assume that the points $v_1,...,v_i$ have already been coloured ($i<n$) such that no edge is monochromatic. Now, have a look at $v_{i+1}$. If we cannot colour it in black then this is because of an edge $E_1\subseteq\{v_1,...,v_{i+1}\}$ with $v_{i+1}\in E_1$, such that $E_1\cap\{v_1,...,v_i\}$ is monochromatically black. If we cannot colour it in white then this is because of an edge $E_2\subseteq\{v_1,...,v_{i+1}\}$ with $v_{i+1}\in E_2$, such that $E_2\cap\{v_1,...,v_i\}$ is monochromatically white. Both situations cannot hold at the same time (why?). Thus, we can indeed colour $v_{i+1}$ without violating the requirement, etc.