KKM—A Topological Approach For Trees

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 TR (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.

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Correspondence to Eli Berger.

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Berger, E. KKM—A Topological Approach For Trees. Combinatorica 25, 1–18 (2004). https://doi.org/10.1007/s00493-005-0001-y

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Mathematics Subject Classification (2000):

  • 05B40