5,402 research outputs found
Ramsey numbers for partially-ordered sets
We present a refinement of Ramsey numbers by considering graphs with a
partial ordering on their vertices. This is a natural extension of the ordered
Ramsey numbers. We formalize situations in which we can use arbitrary families
of partially-ordered sets to form host graphs for Ramsey problems. We explore
connections to well studied Tur\'an-type problems in partially-ordered sets,
particularly those in the Boolean lattice. We find a strong difference between
Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the
partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl
Poset-free Families and Lubell-boundedness
Given a finite poset , we consider the largest size \lanp of a family
\F of subsets of that contains no subposet . This
continues the study of the asymptotic growth of \lanp; it has been
conjectured that for all , \pi(P):= \lim_{n\rightarrow\infty} \lanp/\nchn
exists and equals a certain integer, . While this is known to be true for
paths, and several more general families of posets, for the simple diamond
poset \D_2, the existence of frustratingly remains open. Here we
develop theory to show that exists and equals the conjectured value
for many new posets . We introduce a hierarchy of properties for
posets, each of which implies , and some implying more precise
information about \lanp. The properties relate to the Lubell function of a
family \F of subsets, which is the average number of times a random full
chain meets \F. We present an array of examples and constructions that
possess the properties
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
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