State the Mayer-Vietoris theorem for a simplicial complex $K$ which is the union of two subcomplexes $M$ and $N$. Explain briefly how the connecting homomorphism $\partial_{n}: H_{n}(K) \rightarrow H_{n-1}(M \cap N)$ is defined.

If $K$ is the union of subcomplexes $M_{1}, M_{2}, \ldots, M_{n}$, with $n \geqslant 2$, such that each intersection

$$M_{i_{1}} \cap M_{i_{2}} \cap \cdots \cap M_{i_{k}}, \quad 1 \leqslant k \leqslant n,$$

is either empty or has the homology of a point, then show that

$$H_{i}(K)=0 \quad \text { for } \quad i \geqslant n-1 .$$

Construct examples for each $n \geqslant 2$ showing that this is sharp.