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.

中文翻译：

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r 元组的 s 递增序列的长度

我们证明了许多与 Po-Shen Loh [9] 的问题相关的结果，这相当于 Ramsey 理论中的一个问题。让一种= (一种1,一种2,一种3） 和b= (b1,b2,b3) 是整数的两个三元组。定义一种成为 2- 小于 b如果一种一世<b一世对于至少两个值一世, 并定义一个序列一种1, …,一种米三元组为 2- 增加如果一种r是 2-小于一种s每当r<s. Loh 询问如果所有三元组都取 {1, 2, ...,n}，并给出一个日志*对平凡上界的改进n2通过使用三角形去除引理。在另一个方向，一个简单的结构给出了一个下限n3/2. 我们研究这个问题和一系列概括，改进一些已知界限，指出与极值组合学中其他众所周知的问题的联系，并提出一些进一步的问题。