An Extension of Ramsey\u27s Theorem to Multipartite Graphs

Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey\u27s Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number

Ordering infinite utility streams comes at the cost of a non-Ramsey set.

The existence of a Paretian and finitely anonymous ordering in the set of infnite utility streams implies the existence of a non-Ramsey set (a nonconstructive object whose existence requires the axiom of choice). Therefore, each Paretian and finitely anonymous quasi-ordering either is incomplete or does not have an explicit description. Hence, the possibility results of Svensson (1980) and of Bossert, Sprumont, and Suzumura (2006) do require the axiom of choice.Intergenerational justice; Pareto; Multi-period social choice; Axiom of choice; Constructivism;

Ordering infinite utility streams comes at the cost of a non-Ramsey set

The existence of a Paretian and finitely anonymous ordering in the set of infinite utility streams implies the existence of a non-Ramsey set (a nonconstructive object whose existence requires the axiom of choice). Therefore, each Paretian and finitely anonymous quasi-ordering either is incomplete or does not have an explicit description. Hence, the possibility results of Svensson (1980) and of Bossert, Sprumont, and Suzumura (2006) do require the axiom of choice.Intergenerational justice; Pareto; Multi-period social choice; Axiom of choice; Constructivism.

Superfilters, Ramsey theory, and van der Waerden's Theorem

Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Ko\v{c}inac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.Comment: Among other things, the results of this paper imply (using its one-dimensional version) a higher-dimensional version of the Green-Tao Theorem on arithmetic progressions in the primes. The bibliography is now update

On shifted products which are powers

In this note, we improve upon results of Bugeaud, Gyarmati, Sarkozy and Stewart concerning the size of a subset A of {1,...,N} such that the product of any two distinct elements of A plus 1 is a perfect power. We also show that the cardinality of such a set is uniformly bounded assuming the ABC-conjecture, thus improving upon a result of Dietmann, Elsholtz, Gyarmati and Simonovits

Two extensions of Ramsey's theorem

Ramsey's theorem, in the version of Erd\H{o}s and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,...,n} contains a monochromatic clique of order 1/2\log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of R\"odl, we show that there is a constant $c>0$ such that every 2-coloring of the edges of the complete graph on \{2, 3,...,n\} contains a monochromatic clique S for which the sum of 1/\log i over all vertices i \in S is at least c\log\log\log n. This is tight up to the constant factor c and answers a question of Erd\H{o}s from 1981. Motivated by a problem in model theory, V\"a\"an\"anen asked whether for every k there is an n such that the following holds. For every permutation \pi of 1,...,k-1, every 2-coloring of the edges of the complete graph on {1, 2, ..., n} contains a monochromatic clique a_1<...<a_k with a_{\pi(1)+1}-a_{\pi(1)}>a_{\pi(2)+1}-a_{\pi(2)}>...>a_{\pi(k-1)+1}-a_{\pi(k-1)}. That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erd\H{o}s, Hajnal and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.Comment: 21 pages, accepted for publication in Duke Math.

Scholarship on the relations between Ludwig Wittgenstein and Charles S. Peirce

Thirty years ago Richard Rorty detected the similarities between Wittgenstein's Philosophical Investigations (1953) and the philosophical framework of Charles S. Peirce (1839-1914), the founder of pragmatism. Rorty tried to show that Peirce envisaged and repudiated in advance logical positivism and developed insights and a philosophical mood very close to the analytical philosophers influenced by the later Wittgenstein (Rorty 1961). In spite of that, the majority of scholars have considered both thinkers as totally alien. Some scholars have attributed the pragmatist flavor of the Philosophical Investigations to the influence of Frank P. Ramsey, who awoke Wittgenstein from the dogmatic slumber of the Tractatus. Nevertheless, the real scope of the influence of American pragmatist philosophy in Wittgenstein's later thought is not clearly known. The purpose of my paper is not to describe the common themes between Wittgenstein and Peirce, but the way in which recent scholarship has established some links between both philosophers

Corruption and trade tariffs, or a case for uniform tariffs

By explicitly accounting for the interaction between importers and corrupt customs officials, the author argues that setting trade tariff rates at a uniform level, limits public official's ability to extract bribes from importers. If the government's main objective is to raise revenues at the minimum cost to welfare, optimally-set tariff rates will be inversely proportional to the elasticity of demand for imports. So they will generally differ across goods. Such a menu of tariff rates endows customs officials with the opportunity to extract rent from importers. If officials have enough discretionary power, they might threaten to misclassify goods into more heavily taxed categories unless importers pay them a bribe. Because of the bribe, the effective tariff rate for the importing firm increases, so demand for the good decreases. The resulting drop in import demand implies an efficiency loss as well as lower government revenues, compared with the optimal taxation benchmark without corruption. A similar argument applies when customs officials offer to classify goods into low-tariff categories in exchange for a bribe. Setting trade tariffs at a uniform level eliminates officials'opportunities to extract rents. Thus, when corruption is pervasive, a uniform tariff can deliver more government revenues and welfare than the optimally set (Ramsey) tariff benchmark. The empirical evidence confirms that these considerations are relevant to policymaking, since a robust association between the standard deviation of trade tariffs - a measure of the diversification of tariff menus - and corruption emerges across countries.Trade Policy,Environmental Economics&Policies,Economic Theory&Research,Export Competitiveness,Decentralization,Environmental Economics&Policies,TF054105-DONOR FUNDED OPERATION ADMINISTRATION FEE INCOME AND EXPENSE ACCOUNT,Economic Theory&Research,Export Competitiveness,Trade Policy