6,844 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
Diamond-free Families
Given a finite poset P, we consider the largest size La(n,P) of a family of
subsets of that contains no subposet P. This problem has
been studied intensively in recent years, and it is conjectured that exists for general posets P,
and, moreover, it is an integer. For let \D_k denote the -diamond
poset . We study the average number of times a random
full chain meets a -free family, called the Lubell function, and use it for
P=\D_k to determine \pi(\D_k) for infinitely many values . A stubborn
open problem is to show that \pi(\D_2)=2; here we make progress by proving
\pi(\D_2)\le 2 3/11 (if it exists).Comment: 16 page
On Quasiminimal Excellent Classes
A careful exposition of Zilber's quasiminimal excellent classes and their
categoricity is given, leading to two new results: the L_w1,w(Q)-definability
assumption may be dropped, and each class is determined by its model of
dimension aleph_0.Comment: 16 pages. v3: correction to the statement of corollary 5.
An improvement of the general bound on the largest family of subsets avoiding a subposet
Let be the maximum size of a family of subsets of not containing as a (weak) subposet, and let be the length of
a longest chain in . The best known upper bound for in terms of
and is due to Chen and Li, who showed that for any fixed .
In this paper we show that for any fixed , improving the best known upper bound. By choosing appropriately, we
obtain that as a corollary, which we show is best
possible for general . We also give a different proof of this corollary by
using bounds for generalized diamonds. We also show that the Lubell function of
a family of subsets of not containing as an induced subposet is
for every .Comment: Corrected mistakes, improved the writing. Also added a result about
the Lubell function with forbidden induced subposets. The final publication
will be available at Springer via http://dx.doi.org/10.1007/s11083-016-9390-
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