Ladders in ordered sets (Q1080872)

A partially ordered set (or, shortly, ordered set) P is said to have the 2-cut property if for every element x of P there is a subset X of P whose elements are noncomparable to x such that \(| X| \leq 2\) and such that every maximal chain in P meets \(\{x\}\cup X\). An ordered set P is called a ladder of length n, if \(P=\cup (\{x_ i,y_ i,z_ i\}\); \(i=1,...,n)\), \(x_ 1<x_ 2<...<x_ n\), \(z_ 1<z_ 2<...<z_ n\) and \(x_ i<y_ i<z_ i\) for every \(i=1,...,n\). The main result: Let P be an ordered set with 2-cut property and width n. Then P contains a ladder of length \([1/2(n-3)].\)