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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)
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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

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