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.



Poset dimension and width (Dilworth's theorem)
For your modified question, namely, does there exist a poset $P$ such that $\dim(P) > \sup \{|A|: A\subset P \text{ is an antichain}\}$, the answer is positive. Due to Laver (*An Order Type Decomposition Theorem, Annals of Mathematics Second Series, Vol. 98, No. 1 (Jul., 1973), pp. 96-119*) and in fact he showed that 



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> **for any uncountable $\kappa$, there exist a poset $P$ whose width is $\aleph\_0$ and dimension is $\kappa$.**
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The poset is the class of scattered linear orders of size $<\kappa$ ordered by embeddability (modulo bi-embeddability to make it a poset).


This in some sense is a strengthening of Perles' result.


