We give a simple constructive version of Szemerédi’s Regularity Lemma, based on the computation of singular values of matrices. Mathematical Reviews Subject Numbers: 05C85, 68R10.

The notion of ultrametrics can be considered as a zero-dimensional analogue ordinary metrics, and it is expected to prove ultrametric versions theorems on metric spaces. In this paper, we provide the Arens-Eells isometric embedding theorem spaces, Hausdorff extension Niemytzki-Tychonoff characterization compactness, author’s interpolation metrics dense subsets spaces metrics.

In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:Xtimes Yto Z$, depended on  a natural number $n$ and a cardinal number $kappa$; which is called $n$-factorization property of level $kappa$. Then we study the relation between $n$-factorization property of  level $kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity...

R. H. Schelp conjectured that if G is a graph with |V (G)| = R(Pn, Pn) such that δ(G) > 3|V (G)| 4 , then in every 2-colouring of the edges of G there is a monochromatic Pn. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree. Here we prove Ramsey-type results that imply the conjecture in a weakened form, first rep...

The notions of fuzzy bi-ideals and fuzzy quasi-ideals in ordered semirings are introduced and some of their characterizations are obtained through regularity criterion.

We give a stability-theoretic proof of the algebraic regularity lemma from [6], in a slightly strengthened form. We also point out that the underlying lemmas hold at a greater level of generality, namely “measurable” theories and structures in the sense of Elwes-MacphersonSteinhorn.

The notion of left convergent sequences of graphs introduced by Lovász et al. (in relation with homomorphism densities for fixed patterns and Szemerédi’s regularity lemma) got increasingly studied over the past 10 years. Recently, Nešetřil and Ossona de Mendez introduced a general framework for convergence of sequences of structures. In particular, the authors introduced the notion of QF -conve...