Some cofinality theorems on ordered sets

On changing cofinality of partially ordered sets

It is shown that under GCH every poset preserves its cofinality in any cofinality preserving extension. On the other hand, starting with ω measurable cardinals, a model with a partial ordered set which can change its cofinality in a cofinality preserving extension is constructed.

Some Representation Theorems for Partially Ordered Sets

Some intersection theorems for ordered sets and graphs

Of particular interest are those B for which the maximum families consist of socalled “kernel systems,” i.e., the family of all supersets of some fixed set in B. For example, we show that the set of all (cyclic) translates of a block of consecutive integers in [n] is such a family. It turns out rather unexpectedly that many of the results we obtain here depend strongly on properties of the well...

Antichains 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 ...

Perfect Codes on Some Ordered Sets

Using the concept of codes on ordered sets introduced by Brualdi, Graves and Lawrence, we consider perfect codes on the ordinal sum of two ordered sets, the standard ordered sets and the disjoint sum of two chains. In the classical coding theory all perfect codes are completely described in terms of parameters (cf. [2, 3]). They are mathematically interesting structures. Brualdi et al. [1] rece...