125 research outputs found
Weak embeddings of posets to the Boolean lattice
The goal of this paper is to prove that several variants of deciding whether
a poset can be (weakly) embedded into a small Boolean lattice, or to a few
consecutive levels of a Boolean lattice, are NP-complete, answering a question
of Griggs and of Patkos. As an equivalent reformulation of one of these
problems, we also derive that it is NP-complete to decide whether a given graph
can be embedded to the two middle levels of some hypercube
More on Decomposing Coverings by Octants
In this note we improve our upper bound given earlier by showing that every
9-fold covering of a point set in the space by finitely many translates of an
octant decomposes into two coverings, and our lower bound by a construction for
a 4-fold covering that does not decompose into two coverings. The same bounds
also hold for coverings of points in by finitely many homothets or
translates of a triangle. We also prove that certain dynamic interval coloring
problems are equivalent to the above question
Note on polychromatic coloring of hereditary hypergraph families
We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but
all its restricted subhypergraphs with edges of size at least 3 are
2-colorable. This disproves a bold conjecture of Keszegh and the author, and
can be considered as the first step to understand polychromatic colorings of
hereditary hypergraph families better since the seminal work of Berge. We also
show that our method cannot give hypergraphs of arbitrary high uniformity, and
mention some connections to panchromatic colorings
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