On a scale of criteria on $n$-dependence

In this paper we prove that a planar set $\mathcal{X}$ of at most $mn-1$ points, where $m\le n$, is $\kappa$-dependent, if and only if there exists a number r, $1\le r\le m-1$, and an essentially $\kappa$-dependent subset $\mathcal{Y}\subset \mathcal{X}$, $\mathrm{\#}\mathcal{Y}\ge rs$, where $r + s - 3 = \kappa$,belongingtoanalgebraiccurveofdegree$r$, and not belonging to any curve of degree less than $r$. Moreover, if $\mathrm{\#}\mathcal{Y}=rs$ then the set $\mathcal{Y}$ coincides with the set of intersection points of some two curves of degrees $r$ and $s$, respectively. Let us mention that the first three criteria of the scale, for $m=1,2,3,$ are well-known results.