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Let the set of nodesXin the plain ben-independent, i.e., each node has afundamental polynomial of degreen.Suppose also that|X|= (n+1)+n+···++(n−k+4)+2 and 3≤k≤n−1.We prove that there can be at most 4 line-arly independent curves of degree less than or equal tokpassing through all thenodes ofX.We provide a characterization of the case when there are exactly 4such curves. Namely, we prove that then the setXhas a very special construc-tion: all its nodes but two belong to a (maximal) curve of degreek−2.At theend, an important application to the Gasca-Maeztu conjecture is provided.