Finite union theorem with restrictions

Abstract

The aim of this paper is to prove the following extension of the Folkman-Rado-Sanders Finite Union Theorem: For every positive integersr andk there exists a familyL of sets having the following properties: i) ifS1,S2, ...,Sk + 1 are distinct pariwise disjoint elements ofL then there exists nonemptyI ⊂ {1, 2, ...,k + 1} with ∪i∈ISi⋃L ii) ifL =L1 ⋃...⋃Lr is an arbitrary partition then there existsj ≤ r and pairwise disjoint setsS1,S2, ...,Sk∈Lj, such thatLi∈ISi∈Lj for every nonemptyI ⊂ {1, 2, ...,k}.