$2$-colourability of the maximum ranked elements of a combinatorially sphere-like ranked poset

We obtain a higher dimensional analogue of a classical theorem which states that a polygonally cellulated $2$-sphere in $\mathbb{R}^3$, such that each vertex has even degree, is $2$-face-colourable. In order to formulate our result, we introduce the notion of combinatorially sphere-like ranked posets, which are ranked posets that generalise combinatorial spheres. We prove that, in a combinatorially sphere-like ranked poset $S$ of rank $k$, if each element of rank $(k-2)$ is covered by an even number of elements, then the maximum ranked elements of $S$ admit a proper $2$-colouring, i.e., any two adjacent maximum ranked elements have different colours.