In a seminal paper from 1983, Burr and Erd1⁄2os started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various clas...

The Szemerédi Regularity Lemma states that any sufficiently large graph G can be partitioned into a bounded (independent of the size of the graph) number of regular, or “random-looking,” components. The resulting partition can be viewed as a regularity graph R. The Key Lemma shows that under certain conditions, the existence of a subgraph H in R implies its existence in G. We prove the Regulari...

*Correspondence: [email protected] 2Otto-von-Guericke Universität, Magdeburg, Germany Full list of author information is available at the end of the article Abstract We describe an algorithm which finds binomials in a given ideal I ⊂ Q[x1, . . . , xn] and in particular decides whether binomials exist in I at all. Binomials in polynomial ideals can be well hidden. For example, the lowest degr...

We study the relationship between the Tor-regularity and the local-regularity over a positively graded algebra defined over a field which coincide if the algebra is a standard graded polynomial ring. In this case both are characterizations of the so-called Castelnuovo–Mumford regularity. Moreover, we can characterize a standard graded polynomial ring as a K-algebra with extremal properties with...

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomol-ogy. We establish the key properties of regularity: its connection with the ...

We use the theory of graph limits to study several quasirandom properties, mainly dealing with various versions of hereditary subgraph counts. The main idea is to transfer the properties of (sequences of) graphs to properties of graphons, and to show that the resulting graphon properties only can be satisfied by constant graphons. These quasi-random properties have been studied before by other ...

In 1998 Luczak, Rödl and Szemerédi [7] proved, by means of the Regularity Lemma, that there exists n0 such that, for any n ≥ n0 and two-edge-colouring of Kn, there exists a pair of vertex disjoint monochromatic cycles of opposite colours covering the vertices of Kn. In this paper we make use of an alternative method of finding useful structure in a graph, leading to a proof of the same result w...

According to a theorem of Szemerédi (1975), for every positive integer k and every δ > 0, there exists S = S(k, δ) such that every subset X of {1, 2, . . . , S} of size at least δS contains an arithmetic progression with k terms. Here we generalize the above result to any set of real numbers. For every positive integer k and every c > 0, there exists N = N(k, c) such that every set of n ≥ N poi...

A theorem of Green, Tao, and Ziegler can be stated as follows: if R is a pseudorandom distribution, and D is a dense distribution of R, then D can be modeled as a distribution M which is dense in uniform distribution such that D and M are indistinguishable. The reduction involved in the proof has exponential loss in the distinguishing probability. Reingold et al give a new proof of the theorem ...

Frequency of firing and regularity of discharge of human motor units, and short term synchrony between pairs of motor units, have been assessed in extensor digitorum communis (EDC) and tibialis anterior (TA) muscles in control subjects and in clinically complete paraplegic subjects. The discharge pattern of TA motor units in paraplegia ranged from extremely regular to very irregular for differe...