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Analogous to the fixed point property for ordered sets, a graph has the fixed vertex property iff each of its endomorphisms has a fixed vertex. The fixed point theory for ordered sets can be embedded into the fixed vertex theory for graphs. Therefore, the potential for cross-fertilization should be explored.
Recall that in a laminar family, any two sets are either disjoint or contained one in the other. Here, a parametrized weakening of this condition is introduced. Let us say that a set system F ⊆ 2 X is t-laminar if A , B ∈ F with | A ∩ B | ≥ t implies A ⊆ B or B ⊆ A ...
Let ℬ n be the poset generated by the subsets of [n] with the inclusion relation and let P be a finite poset. We want to embed P into ℬ n as many times as possible such that the subsets in different copies are incomparable. The maximum number of such embeddings is asymptotically determined for...
The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with Δ(F)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ−j and d≥2j+2 for 0≤j≤Δ−1. This both improves and generalizes the result for trees in Dunn, C. (Discret. Math...
We are interested in maximizing the number of pairwise unrelated copies of a poset P in the family of all subsets of [n]. For instance, Sperner showed that when P is one element, n ⌊ n 2 ⌋ is the maximum number of copies of P. Griggs, Stahl, and Trotter have shown that when P is a chain on k elements, 1 2 k − 1 ...
It is known that the set of all simple graphs is not well-quasi-ordered by the induced subgraph relation, i.e. it contains infinite antichains (sets of incomparable elements) with respect to this relation. However, some particular graph classes are well-quasi-ordered by induced subgraphs. Moreover, some of them are well-quasi-ordered by a stronger relation called labelled induced subgraphs. In this...
The quantisation of the Boolean algebra 2 is given by the semi-integral regularization of the quantale of all join preserving self-maps of the chain of three elements. On this basis prime elements of quantales are identified with strong homomorphisms taking their values in the quantisation of 2. A quantale is spatial iff strong homomorphisms with values in the quantisation of 2 separate elements....
A topological space has the fixed point property if every continuous self-map of that space has at least one fixed point. We demonstrate that there are serious restraints imposed by the requirement that there be a choice of fixed points that is continuous whenever the self-map varies continuously. To even specify the problem, we introduce the universal fixed point property. Our results apply in particular...
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