A direct proof that ℓ∞(3) has generalized roundness zero

The Graph Clustering Problem has a Perfect Zero-Knowledge Proof

Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify several large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual path metric (SSTs...

A Direct Proof That Certain Reduced Products Are Countably Saturated

It is known that reduced products of the following form are ! 1-saturated: i A i =F , where F is a countably incomplete, countably based lter and all A i have the same countable signature. We give an elementary proof of this fact, which avoids reference to Feferman{Vaught's fundamental theorem for products, and quantiier elimination in atomless boolean algebras. JJ onsson and Olin 3] proved tha...

A short proof that every finite graph has a tree-decomposition displaying its tangles

We give a short proof that every finite graph (or matroid) has a tree-decomposition that displays all maximal tangles. This theorem for graphs is a central result of the graph minors project of Robertson and Seymour and the extension to matroids is due to Geelen, Gerards and Whittle.

A simple proof that a word of length n has at most 2n distinct squares

Fraenkel and Simpson [2] proved that the number of distinct squares in a word of length n is at most 2n. Their proof uses a rather intricate combinatorial result of Crochemore and Rytter [1] concerning the lengths of three squares which are prefixes of each other. The same proof is included also in Lothaire’s second book [4, p.281-2]. We give here a very short direct proof of this important res...