Regularity and Positional Games

  title={Regularity and Positional Games},
  author={Alfred W. Hales and Robert I. Jewett},
  journal={Transactions of the American Mathematical Society},
  • A. Hales, R. Jewett
  • Published 1 February 1963
  • Mathematics
  • Transactions of the American Mathematical Society
1.Introduction. Suppose X is a set, π’ž a collection of sets (usually subsets of X), and N is cardinal number. Following the terminology of Rado [1], we say π’ž is N-regular in X if,for any partition of X into N parts, some part has as a subset a member of π’ž. if π’ž is n-regular in X for each integer n, we say π’ž is regular in X.Β 

Euclidean Ramsey Theorems I

A Ramsey-Type Theorem in the Plane

We show that, for any finite set P of points in the plane and for any integer k β‰₯ 2, there is a finite set R = R(P, k) with the following property: for any k-colouring of R there is a monochromatic…

Induced lines in Hales-Jewett cubes

Degrees in oriented hypergraphs and sparse Ramsey theory

Ramsey games

Dense Subsets of Products of Finite Trees

We prove a β€œuniform” version of the finite density Halpern–Lauchli theorem. Specifically, we say that a tree T is homogeneous if it is uniquely rooted and there is an integer , called the branching…

Ramsey’s theorem for $n$-parameter sets

Classes of objects called Β«-parameter sets are defined. A Ramsey theorem is proved to the effect that any partitioning into r classes of the Β»c-parameter subsets of any sufficiently large Β«-parameter…

A study on some combinatorial sets in partial semigroups

  • Arpita Ghosh
  • Mathematics
    Asian-European Journal of Mathematics
  • 2022
In this article, we investigate the image and preimage of the important combinatorial sets such as central sets, $C$-sets, and $J_\delta$-sets which play an important role in the study of…



Axiomatic Treatment of Rank in Infinite Sets

  • R. Rado
  • Mathematics
    Canadian Journal of Mathematics
  • 1949
1. Summary. In many branches of mathematics the notion of rank plays an important part. H. Whitney [3] made a detailed axiomatic investigation of rank and several related ideas. All sets considered…

Choice functions and Tychonoff’s theorem

Theorem. Let iXa\aET) be a family of finite sets, let zA be the class of all finite subsets of I, and for each A E&f let 4>A be a choice function of iXa\aEA). Then there exists a choice function <t>…

Khinchin, Three pearls of number

  • 1952

On Representatives of Subsets

Let a set S of mn things be divided into m classes of n things each in two distinct ways, (a) and (b); so that there are m (a)-classes and m (b)-classes. Then it is always possible to find a set R of…

and M

  • A. Girshick, Theory of games and statistical decisions, Wiley, New York,
  • 1954

Regularity. One of the first problems

  • LEMMA 1. Let M and N be cardinal numbers
  • 1927