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...

‎let $x$‎, ‎$y$ and $z$ be banach spaces and $f:xtimes y‎ ‎longrightarrow z$ a bounded bilinear map‎. ‎in this paper we‎ ‎study the relation between arens regularity of $f$ and the‎ ‎reflexivity of $y$‎. ‎we also give some conditions under which the‎ ‎arens regularity of a banach algebra $a$ implies the arens‎ ‎regularity of certain banach right module action of $a$‎ .

A bipartite graph G = (V1 ∪ V2, E) is (δ, d)-regular if ̨̨ d− d(V ′ 1 , V ′ 2 ) ̨̨ < δ whenever V ′ i ⊂ Vi, |V ′ i | ≥ δ|Vi|, i = 1, 2.Here, d(V ′ 1 , V ′ 2 ) = e(V ′ 1 , V ′ 2 )/|V ′ 1 ||V ′ 2 | stands for the density of the pair (V ′ 1 , V ′ 2 ). An easy counting argument shows that if G = (V1 ∪ V2 ∪ V3, E) is a 3-partite graph whose restrictions on V1 ∪V2, V1 ∪V3, V2 ∪V3 are (δ, d)regular, then ...

The new class of Banach spaces, so-called asymptotic lp spaces, is introduced and it is shown that every Banach space with bounded distortions contains a subspace from this class. The proof is based on an investigation of certain functions, called enveloping functions, which are intimately connected with stabilization properties of the norm. 0 Introduction During the last year several problems ...

Dave Benson, in [2], conjectured that for any finite group G and any prime p the Castelnuovo-Mumford regularity of the cohomology ring, H(G,Fp), is zero. He showed that reg(H(G,Fp)) ≥ 0 and succeeded in proving equality when the difference between the dimension and the depth is at most two. The purpose of this paper is to prove Benson’s Regularity Conjecture as a corollary of the following result.

A celebrated result of Gowers states that for every > 0 there is a graph G such that every -regular partition of G (in the sense of Szemerédi’s regularity lemma) has order given by a tower of exponents of height polynomial in 1/ . In this note we give a new proof of this result that uses a construction and proof of correctness that are significantly simpler and shorter.