Theorem 1. For any integers r, d>1 there exists an integer T=T(d, r), such that given sets A 1 , ..., A d+1 /R d in general position, consisting of T points each, one can find disjoint (d+1)-point sets S 1 , ..., S r such that each S i contains exactly one point of each A j , j=1, 2, ..., d+1, and the simplices spanned by S 1 , ..., S r all have a point in common.
This was conjectured by Ba ra ny, Fu redi, and Lova sz [2] in the context of a combinatorial geometry problem (of so-called k-sets, see also [1]), and it was proved by Z 2 ivaljevic and Vrec ica [8] by a topological method. Their proof establishes a suitable connectivity property of a so-called chessboard complex (also called a complex of injective functions). Bjo rner et al. [4] establish this connectivity by other, simpler means, and they prove a number of other properties of the chessboard complexes.
Here we present yet another proof using a technique developed by Sarkaria [6,7]. This technique allows us to avoid dealing with the connectivity of the chessboard complexes explicitly. All the required more advanced topology is summarized in a single fixed-point type theorem (Theorem 3 below), and the rest of the proof is quite elementary. The proof may still look a bit complicated on the first sight; on the other hand, exactly the same proof scheme applies for proving, e.g., the van Kampen Flores theorem from topology and its generalizations [6] or the celebrated Kneser conjecture and its generalizations [7], and the difference between these proofs is in purely combinatorial considerations, which are in fact quite simple in all the cases mentioned here.
Following the other known proofs, we establish a topological version of Theorem 1: Theorem 2. Let d be a positive integer and p a prime. Let A 1 , ..., A d+1 be disjoint sets of cardinality 2p&1 each. Let K be the (abstract) simplicial article no. 0011 146 0095-8956Â96 12.00