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

TL;DR

This paper investigates the maximum length of sequences of integer triples with a specific increasing property, improving bounds and exploring connections to extremal combinatorics and Ramsey theory.

Contribution

It provides new bounds and insights on the length of s-increasing sequences of r-tuples, extending previous results and connecting to broader combinatorial problems.

Findings

Improved upper bounds using the triangle removal lemma

Constructed lower bounds of n^{3/2} for sequence length

Connected the problem to other extremal combinatorics issues

Abstract

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 $a^1,\dots,a^m$ 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 $\{1,2,\dots,n\}$, and gives a $\log_*$ improvement over the trivial upper 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.