Erdős-Szekeres theorem for multidimensional arrays

Abstract

The classical Erdős-Szekeres theorem dating back almost a hundred years states that any sequence of (n − 1) 2 + 1 distinct real numbers contains a monotone subsequence of length n. This theorem has been generalised to higher dimensions in a variety of ways but perhaps the most natural one was proposed by Fishburn and Graham more than 25 years ago. They defined the concept of a monotone and a lex-monotone array and asked how large an array one needs in order to be able to find a monotone or a lex-monotone subarray of size n × . . . × n. Fishburn and Graham obtained Ackerman-type bounds in both cases. We significantly improve these results. Regardless of the dimension we obtain at most a triple exponential bound in n in the monotone case and a quadruple exponential one in the lex-monotone case. Definition (Monotone array). A d-dimensional arrayone of the following alternatives occurs:(i) f (a 1 , . . . , a i−1 , x, a i+1 , . . . , a d ) is increasing in x for all choices of a 1 , . . . , a i−1 , a i+1 , . . . , a d ;

On quantitative aspects of a canonisation theorem for edge‐orderings

“…Thus the well-known Erdős-Szekeres theorem [11] yields the exact value L (1) (𝑛) = (𝑛 − 1) 2 + 1. Accordingly, Leeb's theorem can be viewed as a multidimensional version of the Erdős-Szekeres theorem and we refer to [1] for further variations on this theme. Our own results can be summarised as follows.…”

“…For 𝑘 = 1, there exist only two canonical orderings of [𝑁] (1) , namely the "increasing" and the "decreasing" one corresponding to the sign vectors 𝜀 = (+1) and 𝜀 = (−1), respectively. Thus the well-known Erdős-Szekeres theorem [11] yields the exact value L (1) (𝑛) = (𝑛 − 1) 2 + 1. Accordingly, Leeb's theorem can be viewed as a multidimensional version of the Erdős-Szekeres theorem and we refer to [1] for further variations on this theme.…”

A Strengthening of the Erdős-Szekeres Theorem

“…The Erdős-Szekeres Theorem [10] states that any sequence of distinct integers of length at least rs + 1 must contain a monotone increasing subsequence of length r + 1 or a monotone decreasing subsequence of length s + 1. This fundamental result in extremal combinatorics has inspired the study of many interesting variations (for example, see [7,11,12,17]). In many of these variations, it is useful to observe that the Erdős-Szekeres Theorem can be interpreted as a statement about ordered graphs.…”

On quantitative aspects of a canonisation theorem for edge-orderings

“…The set Z ‹ " Z tx, x 1 u cannot be impeded for then Z was impeded as well. So by the minimality of Z there exist sets L, R ‹ Ď Z ‹ forming a left comb and a right comb, respectively, and satisfying |L| `|R ‹ | ě 1 2 p|Z ‹ | ´k `1q.…”

“…Thus the well-known Erdős-Szekeres theorem [ 10 ] yields the exact value L p1q pnq " pn ´1q 2 `1. Accordingly, Leeb's theorem can be viewed as a multidimensional version of the Erdős-Szekeres theorem and we refer to [ 1 ] for further variations on this theme. Our own results can be summarised as follows.…”