Graham and Pollak proved that one needs at least n − 1 complete bipartite subgraphs (bicliques) to partition the edge set of the complete graph on n vertices. In this paper, we study the extension of Graham and Pollak’s result to coverings of a graph G where each edge of G is allowed to be covered a specified number of times and its generalization to complete uniform hypergraphs. We also discus...

A famous result of Graham and Pollak states that the complete graph with n vertices can be edge partitioned into n − 1, but no fewer, complete bipartite graphs. This result has led to the study of the relationship between the chromatic and biclique partition numbers of graphs. It has become even more exciting with recent connections to the clique versus stable set problem, communication protoco...

A subset B of an algebra subsets a set Ω has property (N) if each B-pointwise bounded sequence the Banach space ba(A) is in ba(A), where real or complex finitely additive measures defined on endowed with variation norm. (G) [(VHS)] for [if sequence] convergence implies its weak convergence. (sN) [(sG) (sVHS)] every increasing covering {Bn:n∈N} contains Bp [(G) (VHS)], and (wN) [(wG) (wVHS)] web...

We show how to simulate any BPP algorithm in polynomial time by using a weak random source of r bits and min-entropy r for any γ > 0. This follows from a more general result about sampling with weak random sources. Our result matches an information-theoretic lower bound and solves a question that has been open for some years. The previous best results were a polynomial time simulation of RP [M....

The Abstract Compactness Theorem of Makowsky and Shelah for model theoretic logics is shown to be an immediate consequence of a general characterization of topological spaces having [ ; ]-compact products, when applied to spaces of structures endowed with the natural topology induced by the de…nable classes of a logic L. In this context, the notion of an ultra…lter U being related to L correspo...

Borodin, Linial, and Saks introduce a general model for online systems in BLS92] called task systems and show a deterministic algorithm which achieves a competitive ratio of 2n ?1 for any metrical task system with n states. We present a randomized algorithm which achieves a competitive ratio of e=(e?1)n?1=(e?1) 1:5820n?0:5820 for this same problem. For the uniform metric space, Borodin, Linial,...