Codimension Two and Three Kneser Transversals

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Let $k,d,\lambda\geqslant1$ be integers with $d\geqslant\lambda $. In 2010, the following function was introduced: $m(k,d,\lambda)\overset{\mathrm{def}}{=}$ the maximum positive integer $n$ such that every set of $n$ points (not necessarily in general position) in $\mathbb{R}^{d}$ has the property that the convex hulls of all $k$-sets have a common transversal $(d-\lambda)$-plane. This is a continuation of a recent work in which it is introduced and studied a natural discrete version of $m$ by considering the existence of \emph{complete Kneser transversals} (i.e., $(d-\lambda)$-transversals $L$ to the convex hulls of all $k$-sets and $L$ containing $(d-\lambda)+1$ points of the given set of points). In this paper, we introduce and study the notions of \emph{stability} and \emph{unstability}. We give results when $\lambda =2,3$, among other results, we present a classification of (complete) Kneser transversals. These results lead us to new upper and lower bounds for $m$. Finally, by using oriented matroid machinery, we present computational results concerning (complete) Kneser transversal in some special cases.