We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal{T}}_{max}\left(\mathcal{E}\right)$, ${\mathcal{T}}_{min}\left(\mathcal{E}\right)$ and ${\mathcal{T}}_{\mathrm{ort}}\left(\mathcal{E}\right)$. We show that these three configurations are not isolated examples; in any given union-closed set system of infinite breadth, at least one of these three configurations will occur as a subprojection. This characterizes those union-closed set systems which have infinite breadth, and is the first general structural result for such set systems.

中文翻译：

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无限广的联合闭集系统中不可避免的子投影

我们考虑具有无限广度的联合闭集系统，重点关注三种特殊配置 ${\mathcal{Ť}}_{最大限度}（\mathcal{Ë}）$， ${\mathcal{Ť}}_{分}（\mathcal{Ë}）$ 和 ${\mathcal{Ť}}_{\mathrm{奥尔}}（\mathcal{Ë}）$。我们证明这三种配置不是孤立的例子；在任何给定的无限广的联合闭集系统中，这三种配置中的至少一种将作为子投影出现。这表征了具有无限广度的并集封闭系统，并且是此类系统的第一个一般结构结果。