Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-23T13:06:21.320Z Has data issue: false hasContentIssue false
  • English
  • Français

Extremal Problems for Affine Cubes of Integers

Published online by Cambridge University Press:  01 March 1998

DAVID S. GUNDERSON  and
VOJTÉCH RÖDL
DAVID S. GUNDERSON
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (e-mail: triangle@icarus.Math.McMaster.ca and rodl@mathcs.emory.edu)
VOJTÉCH RÖDL
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (e-mail: triangle@icarus.Math.McMaster.ca and rodl@mathcs.emory.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

A collection H of integers is called an affine d-cube if there exist d+1 positive integers x0,x1,…, xd so that

formula here

We address both density and Ramsey-type questions for affine d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of {1,2,…,n} not containing an affine d-cube. In 1892 Hilbert published the first Ramsey-type result for affine d-cubes by showing that, for any positive integers r and d, there exists a least number n=h(d,r) so that, for any r-colouring of {1,2,…,n}, there is a monochromatic affine d-cube. Improvements for upper and lower bounds on h(d,r) are given for d>2.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 7 , Issue 1 , March 1998 , pp. 65 - 79
Copyright
1998 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)