Intersection of linear and convex hulls of some partition of a given \((m+2)\)-point set in \({\mathbb R}^m\) (Q1406452)

The main result of the paper is related with the investigations of \textit{R. Rado} [J. Lond. Math. Soc. 27, 320-328 (1952; Zbl 0047.15703)] and is as follows. For each number \( k\leq m+1 \) the following conditions are equivalent: (1) \(k\geq [m/2]+1\); (2) among any \( m+2 \) points \( P_1,\dots,P_{m+2}\in {\mathbb R}^m \) there exist \(k\) points \( P_{i_1},\dots,P_{i_k} \) such that the plane generated by these points intersects the convex hull of the rest points.