KKM?A Topological Approach For Trees

Abstract

The Knaster–Kuratowski–Mazurkiewicz (KKM) theorem is a powerful tool in many areas of mathematics. In this paper we introduce a version of the KKM theorem for trees and use it to prove several combinatorial theorems. A 2-tree hypergraph is a family of nonempty subsets of T ∪ R (where T and R are trees), each of which has a connected intersection with T and with R. A homogeneous 2-tree hypergraph is a family of subsets of T each of which is the union of two connected sets. For each such hypergraph H we denote by τ (H) the minimal cardinality of a set intersecting all sets in the hypergraph and by ν(H) the maximal number of disjoint sets in it. In this paper we prove that in a 2-tree hypergraph τ(H)≤2ν(H) and in a homogeneous 2-tree hypergraph τ(H)≤3ν(H). This improves the result of Alon [3], that τ(H)≤8ν(H) in both cases. Similar results are proved for d-tree hypergraphs and homogeneous d-tree hypergraphs, which are defined in a similar way. All the results improve the results of Alon [3] and generalize the results of Kaiser [1] for intervals.