This paper concerns the open problem of Lovász and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We first give an example exhibiting the largest gap known. We then prove two related theorems.

It is proved that orthogonal convexity defined by A. JimenezMelado and E. Llorens-F'uster implies the weak Banach-Saks property. Relations between orthogonal convexity and another geometric properties, such as nearly uniform smoothness and property ( P ) , are studied. Introduction. Orthogonal convexity has been introduced by A. Jimenez-Melado and E. Llorens-F'uster (see [3] and [4]) as a geome...

The sensitivity conjecture is a longstanding conjecture concerning the relationship between the degree and sensitivity of a Boolean function. In 2015, a communication game was formulated by Justin Gilmer, Michal Koucký, and Michael Saks to attempt to make progress on this conjecture. Andrew Drucker independently formulated this game. Shortly after the creation of the GKS game, Nisan Szegedy obt...

This paper shows that shared coin algorithms can be combined to optimize several complexity measures, even in the presence of a strong adversary. By combining shared coins of Bracha and Rachman [10] and of Aspnes and Waarts [7], this yields a shared coin algorithm, and hence, a randomized consensus algorithm, with O(n log n) individual work and O(n log n) total work, using single-writer registe...

Saks and West conjectured that for every product of partial orders, the maximum size of a semiantichain equals the minimum number of unichains needed to cover the product. We prove the case where both factors have width 2. We also use the characterization of product graphs that are perfect to prove other special cases, including the case where both factors have height 2. Finally, we make some o...

E. Helly’s theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is o...