# Weighted Rooted Trees: Fat or Tall?

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12159)

## Abstract

Let V be a countable set, let T be a rooted tree on the vertex set V, and let $${\mathcal M}=(V,2^V, \mu )$$ be a finite signed measure space. How can we describe the “shape” of the weighted rooted tree $$(T, {\mathcal M})$$? Is there a natural criterion for calling it “fat” or “tall”? We provide a series of such criteria and show that every “heavy” weighted rooted tree is either fat or tall, as we wish. This leads us to seek hypergraphs such that regardless of how we assign a finite signed measure on their vertex sets, the resulting weighted hypergraphs have either a “heavy” large matching or a “heavy” vertex subset that induces a subhypergraph with small matching number. Here we also must develop an appropriate definition of what it means for a set to be heavy in a signed measure space.

## Keywords

Dilworth’s Theorem Down-set Path

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