By using the Szemerédi Regularity Lemma [10], Alon and Sudakov [1] recently extended the classical Andrásfai-Erdős-Sós theorem [2] to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r+1)-partite graph H whose smallest part has t vertices, there exists a constant C such that for any given ε > 0 and sufficiently large n...

Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n) copies of H can be made H-free by removing o(n) edges. We give a new proof which avoids Szemerédi’s regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers question...

The asymptotics of 2-colour Ramsey numbers of loose and tight cycles in 3-uniform hypergraphs were recently determined [16, 17]. We address the same problem for Berge cycles and for 3 colours. Our main result is that the 3-colour Ramsey number of a 3-uniform Berge cycle of length n is asymptotic to 5n 4 . The result is proved with the Regularity Lemma via the existence of a monochromatic connec...

Partitioned symmetric matrices, in particular the Hessian of the Lagrangian, play a fundamental role in nonlinear optimization. For this type of matrices S.-P. Han and 0. Fujiwara recently presented an inertia theorem under a certain regularity assumption. We prove that this theorem is true without any regularity assumption. Then we consider matrix extensions preserving the sign of the determin...

In this paper we unify several existing regularity conditions for graphs, including strong regularity, k-isoregularity, and the t-vertex condition. We develop an algebraic composition/decomposition theory of conditions. Using our theoretical results show that a family non rank 3 graphs known to satisfy 7-vertex condition fulfills even stronger condition, (3,7)-regularity (the notion is defined ...

Recently, Mikołaj Bojańczyk introduced a class of max-regular languages, an extension of regular languages of infinite words preserving many of its usual properties. This new class can be seen as a different way of generalizing the notion of regularity from finite to infinite words. This paper compares regular and max-regular languages in terms of topological complexity. It is proved that up to...

Szemerédi’s Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices.

This vivid term, " connectoplasm " is intended to allude to " cytoplasm " , and means a unstructured, homogeneous mass of neurons. 1 Introduction How can we characterize the dual contributions of biology and environment to language? This is one of the most hotly contested topics in linguistics, psychology, and related disciplines. Even the staunchest adherents to both ends of the " nature / nur...

We prove a hypergraph analogue of the Erdős-Gallai theorem: for any δ > 0 there exists n0 such that for any 0 ≤ α ≤ 1, any k-uniform hypergraph H on n ≥ n0 vertices with at least (α+ δ) ( n k ) edges contains a tight cycle of length at least αn. Moreover, H contains tight cycles of all lengths up to αn which are divisible by k. A key tool in our proof is a ‘regular slice lemma’, which is derive...

Motivated by an Arens regularity problem, we introduce the concepts of matrix Banach space and matrix Banach algebra. The notion of matrix normed space in the sense of Ruan is a special case of our matrix normed system. A matrix Banach algebra is a matrix Banach space with a completely contractive multiplication. We study the structure of matrix Banach spaces and matrix Banach algebras. Then we...