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The length of an s-increasing sequence of r-tuples

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  08 January 2021

W. T. Gowers  and
J. Long
W. T. Gowers
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WB, UK
J. Long*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WB, UK
*
*Corresponding author. Email: jl694@cam.ac.uk
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Abstract

We prove a number of results related to a problem of Po-Shen Loh [9], which is equivalent to a problem in Ramsey theory. Let a = (a1, a2, a3) and b = (b1, b2, b3) be two triples of integers. Define a to be 2-less than b if ai < bi for at least two values of i, and define a sequence a1, …, am of triples to be 2-increasing if ar is 2-less than as whenever r < s. Loh asks how long a 2-increasing sequence can be if all the triples take values in {1, 2, …, n}, and gives a log* improvement over the trivial upper bound of n2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n3/2. We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well-known problems in extremal combinatorics, and asking a number of further questions.

MSC classification

Primary: 05D99: None of the above, but in this section
Secondary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 5 , September 2021 , pp. 686 - 721
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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