A TVERBERG-TYPE RESULT ON MULTICOLORED SIMPLICES

JÁNOS PACH

Abstract. Let P_{1}, P_{2}, …, P_{d+1} be pairwise disjoint n-element point sets in general position in d-space. It is shown that there exist a point O and suitable subsets Q_{i} ⊆ P_{i} (i = 1, 2, …, d + 1) such that |Q_{i}| ≥ c_{d}|P_{i}|, and every d-dimensional simplex with exactly one vertex in each Q_{i} contains O in its interior. Here c_{d} is a positive constant depending only on d.