These are the notes based on the course on Ramsey Theory taught at Universität Hamburg in Summer 2011. The lecture was based on the textbook “Ramsey theory” of Graham, Rothschild, and Spencer [44]. In fact, large part of the material is taken from that book.

neil lyall

Abstract not found

This paper presents an overview of the current state in research directions in the rainbow Ramsey theory. We list results, problems, and conjectures related to existence of rainbow arithmetic progressions in [n] and N. A general perspective on other rainbow Ramsey type problems is given.

Ramsey Theory was a lifelong interest of Paul Erdős. It began [11] in

Given positive integers d and n, there is an integer N such that for every injective map ffrom { 1..... N} a into R there is a subset A = A1 x A2 x... × Ad of { 1..... N} a such that (1) each Aj has n elements, (2) the restriction of f to A is monotone in each coordinate, (3) there is an ordering of the coordinates such that f on A is texicographic with respect to that ordering. Because injection f is Otherwise arbitrary, the direction of monotonicity for each coordinate (increasing or decreasing) and the coordinate ordering for item (3) cannot be prespecified. Results on necessary sizes of N are included.

Abstract. The purpose of this article is to connect the notion of the amenability of a discrete group with a new form of structural Ramsey theory. The Ramsey theoretic reformulation of amenabil-ity constitutes a considerable weakening of the Følner criterion. As a by-product, it will be shown

There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramsey-type theorems to various fields in mathematics are well

Abstract. We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on k-tuples of words (for every natural number k) over a finite alphabet, can be extended to one for partitions on Schreier-type sets of words (of every countable

Ramsey theory is a branch of combinatorics that deals with unavoidable structure in large systems. Euclidean Ramsey theory specifically looks at structure that remains when some geometric object is partitioned, and this note is a survey of some open problems in that field. 1