Large minimal realizers of a partial order

Abstract:

The size of a minimum realizer of a partial order is called the dimension of that partial order. Here we initiate the study of minimal realizers which are not minimum. As an aid to the study of such realizers, we associate to each minimal realizer certain critical digraphs. We characterize all such critical digraphs for the antichain on n elements, and consequently deduce that for $n \geqslant 4$, the maximum size of a minimal realizer is $[{n^2}/4]$.