The length of an s-increasing sequence of r-tuples [article]

W. T. Gowers, J. Long
2016 arXiv   pre-print
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=(a_1,a_2,a_3) and b=(b_1,b_2,b_3) be two triples of integers. Define a to be 2-less than b if a_i<b_i for at least two values of i, and define a sequence a^1,...,a^m of triples to be 2-increasing if a^r is 2-less than a^s 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 _* improvement over the trivial
more » ... pper bound of n^2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n^3/2. 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.
arXiv:1609.08688v2 fatcat:tif7h2xwdffp7iui5eeb3xtvda