A characterisation of posets that are nearly antichains (Q1765956)

Let \(P=(P,\leq )\) be a finite \(n\)-element poset. If \(k\) is a non-negative integer, then a \(k\)-labelling of \(P\) is an injective function \(f:P\rightarrow \{1,\dots,n\}\) such that \(f(a)<f(b)\) if \(a<b\), and \(| f(a)-f(b)| \leq k\) if \(a\) and \(b\) are incomparable. The linear discrepancy ld\((P)\) of \(P\) is the least \(k\) for which there exists a \(k\)-linear labelling of \(P\). Then \(0\leq \text{ld}(P)\leq n-1\), and ld\((P)=0\) iff \(P\) is a chain, and ld\((P)=n-1\) iff \(P\) is an antichain. In the paper, posets \(P\) with ld\((P)=n-2\) are characterized.