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Maximal matchable set in hypergraph with finite edges

Let $n = 2$, and $H$ be the complete bipartite graph with halves $V_1$ a copy of $\mathbb{N}$ and $V_2$ a copy of $\mathbb{R}$. Let $r(M)$ be an element of $V_2$ not covered by a matching $M$. If $M$ ...
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