For a given poset $P$, let $\mathrm{dim}(P)$ denote the least cardinal $\kappa$ such that there exists a $\kappa$-sized collection of linear extensions of $P$, say $\mathcal{L}$, such that $\leq_P = \bigcap_{L\in \mathcal{L}}\leq_L$. Let $\mathrm{width}(P)$ denote the least cardinal $\lambda$ such that every antichain $A\subset P$ has size $<\lambda$.

*Note that the definition of width here is slightly different from the usual definition seen in combinatorics textbooks but we do this in the set-theoretic convention to deal with the possibility that there is no antichain with maximum cardinality*.

Dilworth (https://www.jstor.org/stable/1969503?seq=1#metadata_info_tab_contents) proved that if $\mathrm{width}(P)$ is finite, then $\mathrm{dim}(P)<\mathrm{width}(P)$. The proof goes through the fact that if the width of $P$ is $k$, then $P$ can be decomposed into a union of $<k$ many chains. This fact is not true for $k\geq \aleph_0$. However, the counter-example (Perles' example https://link.springer.com/article/10.1007%2FBF02759806) has dimension 2 so it satisfies $\mathrm{dim}(P)<\mathrm{width}(P)$. Are there examples violating $\mathrm{dim}(P)<\mathrm{width}(P)$?

**EDIT**: Suggested by bof, the width is better defined as the supremum of the sizes of antichains. So with the new definition, Dilworth's theorem states $\mathrm{dim}(P)\leq \mathrm{width}(P)$ for $P$ of finite width and my question will be modified to whether this is true in general.