We characterize ideals of subsets of natural numbers for which some versions of Ramsey’s theorem and Schur’s theorem hold. These are similar to generalizations shown by Frankl, Graham and Rödl in [6]. We show that, in some sense, Ramsey’s theorem, theorem of Bolzano-Weierstrass and “monotone subsequence” theorem characterize the same class of ideals.

The generalized Roper-Suffridge extension operator in Banach spaces is introduced. We prove that this operator preserves the starlikeness on some domains in Banach spaces and does not preserve convexity in some cases. Furthermore, the growth theorem and covering theorem of the corresponding mappings are given. Some results of Roper and Suffridge and Graham et al. in Cn are extended to Banach sp...

An elementary introduction to Maldacena’s AdS/CFT correspondence is given, with some emphasis in the Fefferman-Graham construction. This is based on lectures given by one of us (E.A.) at the Universidad Autonoma de Madrid.

Given a finite set of points P in the Euclidean plane, the Steiner problem asks us to constuct a shortest possible network interconnecting P. Such a network is known as a minimal Steiner tree. The Steiner problem is an intrinsically difficult one, having been shown to be NP-hard [7]; however, it often proves far more tractable if we restrict our attention to points in special geometric configur...

We give a very short proof of the following result of Graham from 1980: For any finite coloring of R, d ≥ 2, and for any α > 0, there is a monochromatic (d + 1)tuple that spans a simplex of volume α. Our proof also yields new estimates on the number A = A(r) defined as the minimum positive value A such that, in any rcoloring of the grid points Z of the plane, there is a monochromatic triangle o...

The Graham-Rothschild Parameter Sets Theorem is one of the most powerful results of Ramsey Theory. (The Hales-Jewett Theorem is its most trivial instance.) Using the algebra of βS, the Stone-Čech compactification of a discrete semigroup, we derive an infinitary extension of the GrahamRothschild Parameter Sets Theorem. Even the simplest finite instance of this extension is a significant extensio...

For r-~2 let p(n, r ) denote the maximum cardinality of a subset A of N = { 1 , 2 . . . . , n} such that there are no B c A and an integer y with S b = y ' . I t is shown that for any e >-0 and bEB n>-n(e), (l+o(l))2~/t'+l>n('-l>/t'+l)~_p(n, r)~_n~§ for all r_~5, and that for every fixed r~_6, p(n,r)=(l+o(1)).21/t'+~)n (~-1)/('§ as n ~ . Let f(n,m) denote t h e m a x i m u m c a r d i n a l i t...

We construct a gravity dual of a boost-invariant flow of an N = 4 SU(N) supersymmetric Yang-Mills gauge theory plasma with chemical potential. We present both a firstorder corrected late-time solution in Eddington-Finkelstein coordinates and a zeroth-order solution in parametric form in Fefferman-Graham coordinates. The resulting background takes the form of a time-dependent AdS Reissner-Nordst...

Several renowned open conjectures in combinatorics and number theory involve arithmetic progressions. Van der Waerden famously proved in 1927 that for each positive integer k there exists a least positive integer w(k) such that any 2-coloring of 1, . . . , w(k) produces a monochromatic k-term arithmetic progression. The best known upper bound for w(k) is due to Gowers and is quite large. Ron Gr...

We apply the method of characteristics for the solution of pde’s to two combinatorial problems. The first is finding an explicit form for a distribution that arises in bioinformatics. The second is a question raised by Graham, Knuth and Patashnik abiout a sequence of generalized binomial coefficients. We find an exact formula, which factors in an interesting way, in the case where one of the si...