A coloured poset is a poset with a unique maximal element that is equipped with a sheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a LeraySerre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homolog...

We study on-line colorings of certain graphs given as intersection graphs of objects “between two lines”, i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axisaligned recta...

We introduce a Morse theory for posets of Bestvina–Brady type combining matchings and height functions. This generalizes Forman's discrete regular CW-complexes extends previous results on $h$ -regular to all finite posets. also develop relative version which allows us compare the topology poset with that given subposet.