Let $k,d,\lambda \geqslant 1$ be integers with $d\geqslant \lambda $ and let $X$ be a finite set of points in $\mathbb{R}^{d}$. A $(d-\lambda)$-plane $L$ transversal to the convex hulls of all $k$-sets of $X$ is called a Kneser transversal. If in addition $L$ contains $(d-\lambda)+1$ points of $X$, then $L$ is called a complete Kneser transversal. In this paper, we present various results on the existence of (complete) Kneser transversals for $\lambda =2,3$. In order to do this, we introduce the notions of stability and instability for (complete) Kneser transversals. We first give a stability result for collections of $d+2(k-\lambda)$ points in $\mathbb{R}^d$ with $k-\lambda\geqslant 2$ and $\lambda =2,3$. We then present a description of Kneser transversals $L$ of collections of $d+2(k-\lambda)$ points in $\mathbb{R}^d$ with $k-\lambda\geqslant 2$ for $\lambda =2,3$. We show that either $L$ is a complete Kneser transversal or it contains $d-2(\lambda-1)$ points and the remaining $2(k-1)$ points of $X$ are matched in $k-1$ pairs in such a way that $L$ intersects the corresponding closed segments determined by them. The latter leads to new upper and lower bounds (in the case when $\lambda =2$ and $3$) for $m(k,d,\lambda)$ defined as the maximum positive integer $n$ such that every set of $n$ points (not necessarily in general position) in $\mathbb{R}^{d}$ admit a Kneser transversal. Finally, by using oriented matroid machinery, we present some computational results (closely related to the stability and unstability notions). We determine the existence of (complete) Kneser transversals for each of the $246$ different order types of configurations of $7$ points in $\mathbb{R}^3$.

  • 1.  J.L. Arocha J. Bracho and  L. Montejano . Ramírez Alfonsín, Transversals to the convex hulls of all $k$-sets of discrete subsets of $\mathbb{R}^{n}$ , J. Combin. Theory Ser. A , 118 ( 2010 ), pp. 197 -- 207 . CrossrefISIGoogle Scholar

  • 2.  A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler, Oriented Matroids, 2nd ed., Encyclopedia Math. Appl., 46, Cambridge University Press, Cambridge, 1999.Google Scholar

  • 3.  B. Bukh J. and  G. Nivasch , Stabbing simplices by points and flats , Discrete Comput. Geom. , 43 ( 2010 ), pp. 321 -- 338 . CrossrefISIGoogle Scholar

  • 4.  B. Bukh and  G. Nivasch , Upper bounds for centerlines , J. Comput. Geom. , 3 ( 2012 ), pp. 20 -- 30 . Google Scholar

  • 5.  J. Chappelon L. Martínez-Sandoval L. Montejano L.P. Montejano J.L. Ramírez Alfonsín and  Complete Kneser , Adv. Appl. Math. , 82 ( 2017 ), pp. 83 -- 101 . CrossrefISIGoogle Scholar

  • 6.  V.L. Dol'nikov , On transversals of families of convex sets, in Research in Theory of Functions of Several Real Variables, Yaroslavl State University, Yaroslavl , Russia , 1981 , pp. 30 -- 36 (in Russian). Google Scholar

  • 7.  L. Finschi, Catalog of Oriented Matroids, http://www.om.math.ethz.ch/Google Scholar

  • 8.  L. Lovász and  Kneser , chromatic number and homotopy , J. Combin. Theory Ser. A , 25 ( 1978 ), pp. 319 -- 324 . CrossrefISIGoogle Scholar

  • 9.  A. Magazinov and A. Pór, An Improvement on the Trivial Lower Bound for the Depth of a Centerline, arXiv:1603.01641v1, 2016.Google Scholar

  • 10.  R. Rado , A theorem on general measure , J. Lond. Math. Soc. , 22 ( 1947 ), pp. 291 -- 300 . Google Scholar

  • 11.  M. Tancer, private communication, 2014--2015.Google Scholar