Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every, if P is a poset and the dimension of a subposet B of P is at most d whenever the cover graph of B is a block of the cover graph of P, then the dimension of P is at most d + 2.We also construct examples that show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.
We assume that the reader is familiar with basic notation and terminology for partially ordered sets (here we use the short term posets), including chains and antichains, minimal and maximal elements, linear extensions, order diagrams, and cover graphs. Extensive background information on the combinatorics of posets can be found in [17, 18].
We will also assume that the reader is familiar with basic concepts of graph theory, including the following terms: connected and disconnected graphs, components, cut vertices, and k-connected graphs for an integer. Recall that when G is a connected graph, a connected induced subgraph H of G is called a block of G when H is 2-connected and there is no subgraph of G which contains H as a proper subgraph and is also 2-connected.
Here are the analogous concepts for posets. A poset P is said to be connected if its cover graph is connected. A subposet B of P is said to be convex if y ∈ B whenever x, z ∈ B and x < y < z in P. Note that when B is a convex subposet of P, the cover graph of B is an induced subgraph of the cover graph of P. A convex subposet B of P is called a component of P when the cover graph of B is a component of the cover graph of P. A convex subposet B of P is called a block of P when the cover graph of B is a block in the cover graph of P.