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together with 3k = 6(√k/2 − 1 + 1)2
6(χ∗(G) + 1)2 completes the proof.
Ck[v]
Acknowledgments **Piotr** **Micek** is supported by Polish National Science Center UMO-2011/03/
D/ST6/01370. Veit Wiechert is supported

In 1977, Trotter and Moore proved that a poset has dimension at most 3 whenever its cover graph is a forest, or equivalently, has treewidth at most 1. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth 3. In this paper we focus on the boundary case of treewidth 2. It was recently shown ...

A family of sets in the plane is simple if the intersection of any subfamily is arc-connected, and it is pierced by a line \(L\) if the intersection of any member with \(L\) is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their ...

We analyze the on-line dimension of partially ordered sets as a value of a two-person game between Algorithm and Spoiler. The game is played in rounds. Spoiler presents an on-line order of width at most w, one point at a time. Algorithm maintains its realizer, i.e., the set of d linear extensions which intersect to the presented order. Algorithm may not change the ordering of the ...

On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents a partially ordered set, point by point. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of width w is a minimum number val(w) such that Algorithm has a strategy using at most ...

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set \(X\) in \(\mathbb{R }^2\) that is not an ...