We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two n-colorings of a graph are said to be orthogonal if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-n squares, ...

3 Polynomial Optimization, Sums of Squares, and Applications 47 Pablo A. Parrilo 3.1 Nonnegative Polynomials and Sums of Squares . . . . . . . . . . 48 3.2 Applications of Sum of Squares Programs . . . . . . . . . . . . . 76 3.3 Special Cases and Structure Exploitation . . . . . . . . . . . . . 86 3.4 Infeasibility Certificates . . . . . . . . . . . . . . . . . . . . . . . 106 3.5 Duality and S...

This paper examines various propositional logics in which the dynamic implication connective ⇒ (discussed in Groenendijk and Stokhof’s (1992) Dynamic Predicate Logic and Kamp’s (1981) Discourse Representation Theory) plays the central role. Our approach is modal: we view ⇒ as a binary modal operator in a two dimensional modal logic. Two dimensional modal logics are systems in which formulas are...

We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1extendibility is incompatible with square; we prove this is close to optimal by showing that 1-extendib...

A latin square of order n possessing a cyclic automorphism of order n is said to be diagonally cyclic because its entries occur in cyclic order down each broken diagonal. More generally, we consider squares possessing any cyclic automorphism. Such squares will be named after Parker, in recognition of his seminal contribution to the study of orthogonal latin squares. Our primary aim is to survey...

Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. In 1998 Fraenkel and Simpson showed that a string of length n contains at most 2n distinct squares. In 2007 Ilie provided an asymptotic upper bound of 2n−Θ(log n). We show that a string of length n contains at most b5n/3c distinct squares. This new upper b...