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Let 885816300604&_mathId=si1.gif&_user=111111111&_pii=S0196885816300604&_rdoc=1&_issn=01968858&md5=92c0cf2f49901eb0bfd2bea6d537aebb" title="Click to view the MathML source">k,d,λ⩾1 be integers with 885816300604&_mathId=si2.gif&_user=111111111&_pii=S0196885816300604&_rdoc=1&_issn=01968858&md5=3e57b826777a68d6d31af52a5daef9bd" title="Click to view the MathML source">d⩾λ. Let 885816300604&_mathId=si3.gif&_user=111111111&_pii=S0196885816300604&_rdoc=1&_issn=01968858&md5=ce431fb952f126b9c482d717585b90e6" title="Click to view the MathML source">m(k,d,λ) be the maximum positive integer n such that every set of n points (not necessarily in general position) in 885816300604&_mathId=si107.gif&_user=111111111&_pii=S0196885816300604&_rdoc=1&_issn=01968858&md5=7d4c8315dace50dcd0a3cb77fd3ee448" title="Click to view the MathML source">Rd has the property that the convex hulls of all k -sets have a common transversal 885816300604&_mathId=si37.gif&_user=111111111&_pii=S0196885816300604&_rdoc=1&_issn=01968858&md5=c013fedb69ad73219b92034efa622bbd" title="Click to view the MathML source">(d−λ)-plane. It turns out that 885816300604&_mathId=si3.gif&_user=111111111&_pii=S0196885816300604&_rdoc=1&_issn=01968858&md5=ce431fb952f126b9c482d717585b90e6" title="Click to view the MathML source">m(k,d,λ) is strongly connected with other interesting problems, for instance, the chromatic number of Kneser hypergraphs and a discrete version of Rado's centerpoint theorem. In the same spirit, we introduce a natural discrete version 885816300604&_mathId=si6.gif&_user=111111111&_pii=S0196885816300604&_rdoc=1&_issn=01968858&md5=ed369363b081ed7fec36478a69566ee1" title="Click to view the MathML source">m⁎ of m by considering the existence of complete Kneser transversals . We study the relation between them and give a number of lower and upper bounds of 885816300604&_mathId=si6.gif&_user=111111111&_pii=S0196885816300604&_rdoc=1&_issn=01968858&md5=ed369363b081ed7fec36478a69566ee1" title="Click to view the MathML source">m⁎ as well as the exact value in some cases. The main ingredient for the proofs are Radon's partition theorem as well as oriented matroids tools. By studying the alternating oriented matroid we obtain the asymptotic behavior of the function 885816300604&_mathId=si6.gif&_user=111111111&_pii=S0196885816300604&_rdoc=1&_issn=01968858&md5=ed369363b081ed7fec36478a69566ee1" title="Click to view the MathML source">m⁎ for the family of cyclic polytopes.