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

@article{Gowers2021TheLO,
title={The length of an s-increasing sequence of r-tuples},
author={William T. Gowers and Jason Long},
journal={Comb. Probab. Comput.},
year={2021},
volume={30},
pages={686-721}
}
• Published 27 September 2016
• Mathematics
• Comb. Probab. Comput.
We prove a number of results related to a problem of Po-Shen Loh , 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…
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