  Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each
node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal
X=d(n,k-3)+3= (n+1)+n+\cdots+(n-k+5)+3$ and $4 \le k\le n-1.$ In this paper we
prove that there are at most seven linearly independent curves of degree less
than or equal to $k$ that pass through all the nodes of $\mathcal X.$ We
provide a characterization of the case when there are exactly seven such
curves. Namely, we prove that then the set $\mathcal X$ has a very special
construction: all its nodes but three belong to a (maximal) curve of degree
$k-3.$ Let us mention that in a series of such results this is the third one.
In the end, an important application to the bivariate polynomial interpolation
is provided, which is essential also for the study of the Gasca-Maeztu
conjecture.
