Skip to main content
×
×
Home

Extremal Problems for Affine Cubes of Integers

  • DAVID S. GUNDERSON (a1) and VOJTÉCH RÖDL (a1)
    • Published online: 01 March 1998
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.

Copyright
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed