Knaster-Kuratowski-Mazurkiewicz (KKM) theorem is a powerful tool in many areas of mathematics. In this talk a version of KKM theorem for trees is introduced and used to prove several combinatorial theorems.

A 2-trees hypergraph is a family of nonempty subsets of the union of two trees T and R, each of which has a connected intersection with T and with R. A homogeneous 2-trees 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 tau(H) the minimal cardinality of a set intersecting all sets in the hypergraph and by nu(H) the maximal number of disjoint sets in it.

In this talk I prove that in a 2-trees hypegraph tau(H) <= 2 nu(H) and in a homogeneous 2-tree hypergraph tau(H) <= 3 nu(H). This improves the result of Alon, that tau(H) <= 8 nu(H) in both cases.