Complete subgraphs of infinite multipartite graphs and antichains in partially ordered sets
نویسندگان
چکیده
منابع مشابه
Complete ordered sets with no infinite antichains
There are several standard constructions by which an arbitrary ordered set P is extended to one in which every subset has an. infimum and a supremuma complete ordered set. The collection I(P) of all initial segments of P ordered by inclusion is one. [A subset I of P is an initial segment if, for x, y E P, x E I whenever x s y and y E I.] I(P) is a complete distributive lattice. AnoWr is the fam...
متن کاملComplete subgraphs in multipartite graphs
Turán’s Theorem states that every graphG of edge density ‖G‖/ (|G| 2 ) > k−2 k−1 contains a complete graph K and describes the unique extremal graphs. We give a similar Theorem for `-partite graphs. For large `, we find the minimal edge density d` , such that every `-partite graph whose parts have pairwise edge density greater than d` contains a K . It turns out that d` = k−2 k−1 for large enou...
متن کاملOrdered sets with no infinite antichains
One approach to analyzing the structure of an ordered set has been to characterize its structure in terms of that of related objects, and vice versa. For example results in [2] characterize in terms of the ordered set P when the lattice Z(P) of lower sets (also called initial segments) of P and the ordered set Id(P) of order ideals (or simply ideals) of P have infinite antichains. The results i...
متن کاملAntichains in partially ordered sets of singular cofinality
In their paper from 1981, Milner and Sauer conjectured that for any poset 〈P,≤〉, if cf(P,≤) = λ > cf(λ) = κ, then P must contain an antichain of size κ. We prove that for λ > cf(λ) = κ, if there exists a cardinal μ < λ such that cov(λ, μ, κ, 2) = λ, then any poset of cofinality λ contains λ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known ...
متن کاملTripled partially ordered sets
In this paper, we introduce tripled partially ordered sets and monotone functions on tripled partiallyordered sets. Some basic properties on these new dened sets are studied and some examples forclarifying are given.
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1986
ISSN: 0012-365X
DOI: 10.1016/0012-365x(86)90069-5