We say that two partial orders on \(\varvec{[n]}\) are compatible if there exists a partial order that refines both of them. This compatibility relation induces a natural set system structure between the collection \(\varvec{\mathcal {F}}\) of all partial orders and the collection \(\varvec{\mathcal {G}}\) of all total orders on \(\varvec{[n]}\) , where each order is identified with the set of orders compatible with it. In this note, we determine the VC-dimension of \(\varvec{\mathcal {F}}\) with respect to \(\varvec{\mathcal {G}}\) , proving that \({\varvec{VC}}_{\varvec{\mathcal {G}}(\mathcal {F}) = \lfloor \frac{n^2}{4}\rfloor }\) for \(\varvec{n \geqslant 4}\) . We also establish bounds on the dual VC-dimension, showing that \(\varvec{2}\varvec{(n-3)} \leqslant {\varvec{VC}}_{\varvec{\mathcal {F}}}(\varvec{\mathcal {G}}) \leqslant n \log _2 \varvec{n}\) for all \(\varvec{n \geqslant 1}\) .
