Let $X$ be an inner product space, let $G$ be a group of orthogonal transformations of $X$, and let $R$ be a bounded $G$-stable subset of $X$. We define very weak and very strong regularity for such pairs $(R,G)$ (in the sense of Szemer\'edi's regularity lemma), and prove that these two properties are equivalent. Moreover, these properties are equivalent to the compactness of the space $(B(H),d_R)/G$. Here $H$ is the completion of $X$ (a Hilbert space), $B(H)$ is the unit ball in $H$, $d_R$ is… Expand

Let $k\leq n$. Each polynomial $p\in\oR[x_1,...,x_n]$ can be uniquely written as $p=\sum_{\mu}\mu p_{\mu}$, where $\mu$ ranges over the set $M$ of all monomials in $\oR[x_1,...,x_k]$ and where… Expand

PTAS's for a much larger class of weighted MAX-rCSP problems which includes as special cases the dense problems and, for r = 2, all metric instances and quasimetric instances; for r > 2, this class includes a generalization of metrics.Expand

Abstract : A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graph… Expand

Abstract.Szemerédi’s regularity lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of… Expand