It has been known for more than 40 years that there are posets with planarcover graphs and arbitrarily large dimension. Recently, Streib and Trotterproved that such posets must have large height. In fact, all knownconstructions of such posets have two large disjoint chains with all points inone chain incomparable with all points in the other. Gutowski and Krawczykconjectured that this feature is necessary. More formally, they conjecturedthat for every $kgeq 1$, there is a constant $d$ such that if $P$ is a posetwith planar cover graph and $P$ excludes $mathbf{k}+mathbf{k}$, then$dim(P)leq d$. We settle their conjecture in the affirmative. The proofinvolves some intermediate results that we believe are of independent interest.
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机译：它已知超过40年的是，具有Planccover图表和任意大维的Posets。最近，Streib和Trotterproved认为这种活页塞必须具有大的高度。实际上，这种存放的所有已知根本都有两个大的不相容链，所有点都不转换，与另一个点相同。 gutowski和krawczykcolpect，这一功能是必要的。更为正式，他们猜测每一笔k geq 1 $，有一个常数$ d $，如果$ p $是posetwith平面盖图和$ p $ supludes $ mathbf {k} + mathbf {k} $，然后$ dim（p） leq d $。我们肯定地解决了他们的猜想。校对透明度我们认为是独立利益的中间结果。
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