Szemerédi’s regularity lemma proved to be a fundamental result in modern graph theory. It had a number of important applications and is a widely used tool in extremal combinatorics. For some applications variants of the regularity lemma were considered. Here we discuss several of those variants and their relation to each other.

sera from ninety one patients clinically suspected to extra intestinal amoebiasis (ex-am) were collected and examined for presence of anti entamoeba histolytica (e.h) antibody. the tests were indirect immunofluorescence antibody, (ifa or iif), gel diffusion (gd) and counter immuno electrophoresis (ciep). thirty one patients (34%) had anti e.h. antibody titres of >1:80. among them eight patients...

in this paper we study optimization problems with infinite many inequality constraints on a banach space where the objective function and the binding constraints are locally lipschitz‎. ‎necessary optimality conditions and regularity conditions are given‎. ‎our approach are based on the michel-penot subdifferential.

in this paper, we investigate the existence of positive solutions for the ellipticequation $delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $omega$ of $r^{n}$, $ngeq2$, with navier boundary conditions. we show that there exists an extremal parameter$lambda^{ast}>0$ such that for $lambda< lambda^{ast}$, the above problem has a regular solution butfor $lambda> lambda^{ast}$, the probl...

This paper presents a methodology for logic synthesis of Boolean functions in the form of regular structures that can be mapped into standard cells or programmable devices. Regularity offers an elegant solution to hard problems arising in layout and test generation, at no extra cost or at the cost of increasing the number of gates, which does not always translate into the increase of circuit ar...

Varying coefficient models have numerous applications in a wide scope of scientific areas. While enjoying nice interpretability, they also allow flexibility in modeling dynamic impacts of the covariates. But, in the new era of big data, it is challenging to select the relevant variables when there are a large number of candidates. Recently several work are focused on this important problem base...

Let F be a fixed graph of chromatic number r + 1. We prove that for all large n the degree sequence of any F -free graph of order n is, in a sense, close to being dominated by the degree sequence of some r-partite graph. We present two different proofs: one goes via the Regularity Lemma and the other uses a more direct counting argument. Although the latter proof is longer, it gives better esti...

Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an `-partite graph with V (G) = V1 ∪ · · · ∪ V` and |Vi| = n for all i ∈ [`], and all pairs (Vi, Vj) are ε-regular of density d for 1 ≤ i < j ≤ ` and ε d, then G contains (1± f`(ε))d “ ` 2 ” × n` cliques K`, where f`(ε) → ...

The aim of this survey is to collect and explain some geometric results whose proof uses graph or hypergraph theory. No attempt has been made to give a complete list of such results. We rather focus on typical and recent examples showing the power and limitations of the method. The topics covered include forbidden configurations, geometric constructions, saturated hypergraphs in geometry, indep...

We introduce a permutation analogue of the celebrated Szemerédi Regularity Lemma, and derive a number of consequences. This tool allows us to provide a structural description of permutations which avoid a specified pattern, a result that permutations which scatter small intervals contain all possible patterns of a given size, a proof that every permutation avoiding a specified pattern has a nea...