aa r X i v : . [ m a t h . C O ] S e p A note on the Erdős-Szekeres theorem in highdimension
Lyuben LichevEcole Normale Supérieure de Lyon, Lyon, FranceSeptember 22, 2020
Abstract
In [4, 7] respectively Burkill and Mirsky, and Kalmanson, prove independently that, for every r ≥ , n ≥ , there is a sequence of r n vectors in R n , which does not contain a subsequence of r + 1 vectors v , v , . . . , v r +1 such that, for every i between 1 and n , ( v ji ) ≤ j ≤ r +1 forms a monotonesequence. Moreover, r n is the largest integer with this property. In this short note, for two vectors u = ( u , u , . . . , u n ) and v = ( v , v , . . . , v n ) in R n , we say that u ≤ v if, for every i between 1 and n , u i ≤ v i . Just like Burkill and Mirsky, and Kalmanson, for every k, ℓ ≥ , d ≥ we find the maximal N , N (which turn out to be equal) such that there are numerical two-dimensional arrays of size ( k + ℓ − × N and ( k + ℓ ) × N , which neither contain a subarray of size k × d , whose columns forma non-decreasing sequence of d vectors in R k , nor contain a subarray of size ℓ × d , whose columns forma non-increasing sequence of d vectors in R ℓ . In a consequent discussion, we consider a generalisationof this setting and make a connection with a famous problem in coding theory. Keywords: Erdős-Szekeres, two-dimensional array, monotone sequence of vectorsMSC Class: 05D10
For every n ≥ , we denote by [ n ] the set of integers between 1 and n . For a sequence ( a i ) ni =1 and any k ∈ [ n ] , a subsequence of ( a i ) ni =1 of length k is any sequence a i , a i , . . . , a i k with ≤ i < i < · · · < i k ≤ n .In this note, an array A of size m × n is a rectangular matrix of size m × n , and a subarray of A is obtainedby intersecting a subset of the rows of A with a subset of the columns of A .The following theorem due to Erdős and Szekeres is one of the cornerstones in the field of Ramseytheory. Theorem 1.1 (Erdős-Szekeres theorem, [5]) . For all integers k, ℓ ≥ , any sequence of kℓ + 1 distinct realnumbers contains either an increasing subsequence of length k + 1 or a decreasing subsequence of length ℓ + 1 . Since the appearance of the theorem in 1935, a vast number of generalisations have been broughtforward. A few of them are [3, 4, 7]. In each of the three papers, a major subject was the topic of findinglarge subarrays of numerical arrays, in which the entries of every row or every column, or both, form amonotone sequence. To be more precise, we state a version of Theorem 1 from [7], which also appears asTheorem 2.4 in [4]:
Theorem 1.2 ([4, 7]) . Let S be a sequence of vectors in R n , ( n ≥ . If S has length at least r n + 1 ,then it contains a subsequence of r + 1 vectors v , v , . . . , v r +1 such that, for every i between 1 and n , ( v ji ) ≤ j ≤ r +1 forms a monotone sequence. Moreover, r n + 1 cannot be replaced by r n .
1n another line of research, the papers [2, 6] consider another notion of monotonicity. For two vectors u = ( u , u , . . . , u n ) and v = ( v , v , . . . , v n ) in R n , we say that u ≤ v if, for every i ∈ [ n ] , u i ≤ v i . Asequence of vectors v , v , . . . , v k ∈ R d is said to be monotone if v ≤ v ≤ · · · ≤ v k (in this case, thesequence is non-decreasing ) or v k ≤ v k − ≤ · · · ≤ v (the sequence is then non-increasing ). An array iscalled monotone if both the set of its rows and the set of its columns form a monotone sequence of vectors.Perhaps the first to introduce this notion were Fishburn and Graham in [6]. The papers [2, 6] treat (ageneralisation in dimension d ≥ of) the problem of finding the minimal N , for which every numericalarray of size N × N contains a monotone subarray of size n × n (they denote this minimal N by M ( n ) ).Our setting interpolates between the weaker notion from [3, 4, 7] and the stronger one from [2, 6]. Inparticular, we will be interested in the order used in [2, 6], but only on the set of columns of an array andnot on the set of rows. We say that a numerical array has property ( k, ℓ, d ) if it contains either a subarray of size k × d , whosecolumns form a non-decreasing sequence of vectors in R k , or a subarray of size ℓ × d , whose columns forma non-increasing sequence of vectors in R ℓ . Theorem 2.1.
For every k, ℓ ≥ , every array of size ( k + ℓ − × (( d − k + ℓk ) + 1) has property ( k, ℓ, d ) .Proof. Our main tool will be the Erdős-Szekeres theorem. We prove the theorem by induction on k + ℓ .The case k + ℓ = 2 is true, since this is the statement of the Erdős-Szekeres theorem itself. Suppose that thestatement is true for every pair of positive integers of sum less than some integer t ≥ . Let k, ℓ be positiveintegers with k + ℓ = t . Consider the first row of an array of size ( k + ℓ − × (( d − k + ℓk ) + 1) . Then, bythe Erdős-Szekeres theorem, either it contains a non-decreasing subsequence of size ( d − k + ℓ − k − ) + 1 , orit contains a non-increasing subsequence of size ( d − k + ℓ − k ) + 1 . In both cases, it is sufficient to considerthe subarray of the remaining t − rows and the columns, whose first entries participate in the monotonesubsequence given above, to conclude by the induction hypothesis. Remark.
For any k, ℓ, M ∈ N , Theorem 2.1 cannot hold for an array of size ( k + ℓ − × M . Indeed, lettingthe first k − rows be equal to , , . . . , M , and the remaining ℓ − rows be equal to M, M − , . . . , ,there is no subarray of the form described in Theorem 2.1. Remark.
In [6] Fishburn and Graham gave an upper bound on M ( n ) of the form towr ( O ( n )) , where thefunction towr k is defined recursively as towr ( n ) = n, towr k ( n ) = 2 towr k − ( n ) . This bound was improvedsignificantly by Bucić, Sudakov and Tran in [2] to M ( n ) ≤ (2 n ) n . Theorem 2.1 may be seen as an improvement of Theorem 2.3 from [2] in the special case when k = ℓ = n .This slightly improves the bound on M ( n ) given there: we obtain M ( n ) ≤ (cid:18) (2 n − + 12 n − (cid:19) ( n − nn ) + 1 . Indeed, consider a numerical array of size ((2 n − + 1) × (cid:18) (cid:18) (2 n − + 12 n − (cid:19) ( n − nn ) + 1 (cid:19) . By theErdős-Szekeres theorem every column of the array contains a monotone subsequence of length n − .Since any such subsequence may be non-decreasing or non-increasing, and the number of possible choicesfor the indices of the rows of its entries is (cid:0) (2 n − +12 n − (cid:1) , one may deduce by the Pigeonhole principle thatthere is a subarray of size (2 n − × (cid:16) ( n − nn ) + 1 (cid:17) , whose rows form a monotone sequence of vectorsof dimension R ( n − ( nn ) +1 . By Theorem 2.1 we find a monotone subarray of size n × n in the latter.2hat is most surprising about Theorem 2.1 is that it is indeed tight. Theorem 2.2.
For every k, ℓ ≥ , there is an array of size ( k + ℓ ) × ( d − k + ℓk ) , which does not haveproperty ( k, ℓ, d ) . Note that Theorem 2.2 is slightly stronger than a purely complementary statement of Theorem 2.1since we consider arrays of size ( k + ℓ ) × ( d − k + ℓk ) and not ( k + ℓ − × ( d − k + ℓk ) .Before constructing an array of size ( k + ℓ ) × ( d − k + ℓk ) , which does not have property ( k, ℓ, d ) , we makean elementary observation. A walk in Z is up-right if it uses only transitions of the type ( x, y ) → ( x + 1 , y ) and ( x, y ) → ( x, y + 1) . Observation 2.3.
For every k, ℓ ∈ N , the number of up-right walks in Z from (0 , to ( k, ℓ ) is (cid:0) k + ℓk (cid:1) . For technical reasons, let us rotate the grid Z at − ◦ and consider walks from (0 , to (cid:16) k + ℓ √ , k − ℓ √ (cid:17) (which is the image of ( k, ℓ ) under the rotation) instead. Each of these walks contains k steps of the type ր and ℓ steps of the type ց . By Observation 2.3 there are (cid:0) k + ℓk (cid:1) such walks, which we list in an arbitraryorder.Let A be an array of size ( k + ℓ ) × (cid:0) k + ℓk (cid:1) with entries in the set {ր , ց} , for which the j − th column ( A i,j ) k + ℓi =1 describes the j − th walk in the list above (one begins the walk with the arrow A ,j ). See Figure 1. √ Figure 1: The figure depicts the example k = 3 , ℓ = 2 . One possibility for the array A is given by ր ր ր ր ր ր ց ց ց ցր ր ր ց ց ց ր ր ր ցր ց ց ր ր ց ր ր ց րց ր ց ր ց ր ր ց ր րց ց ր ց ր ր ց ր ր ր . Now, we show how to relate the array A to the construction of a numerical array, satisfying therequirements of Theorem 2.2. In the beginning, let B be an empty array of size ( k + ℓ ) × ( d − k + ℓk ) . Wewill fill in the entries of B so that each row forms a permutation of the integers from 1 to ( d − k + ℓk ) .For every i between 0 and (cid:0) k + ℓk (cid:1) and for every j ∈ [ k + ℓ ] , divide the j − th row of B into ( d − k + ℓk ) − i groups of ( d − i consecutive entries. For each group, we replace its entries in the j − th row of B withthe consecutive integers in one of the sets [1 , ( d − i ] , [( d − i + 1 , d − i ] , . . . , [( d − k + ℓk ) − ( d − i + 1 , ( d − k + ℓk )] . Since for every i ∈ [ (cid:0) k + ℓk (cid:1) ] we require that the integers from any group of size ( d − i in the j − throw of B are consecutive, we may consider an order relation between the groups, defined by g < g if3very entry in the group g is smaller than every entry in the group g . Now, we will use the j − th rowof the array A to indicate the order of the integers in every group in the j − th row of B , and the orderof the groups as well, as follows. For every row j ∈ [ k + ℓ ] and for every i ∈ [ (cid:0) k + ℓk (cid:1) ] , if A j,i = ր , then forevery group g i containing ( d − i consecutive entries in the j − th row of B , we order the d − groups of ( d − i − consecutive entries, contained in g i , in increasing order. If, on the other hand, A j,i = ց , then forevery group g i of ( d − i entries in the j − th row of B , we order the d − groups of ( d − i − consecutiveentries, contained in g i , in decreasing order. For the sake of clarity, we give an example.For example, consider the case k = 2 , ℓ = 1 , d = 4 . One possibility for the array A is ր ր ցր ց րց ր ր . Since ( d − k + ℓk ) = 3 = 27 , in the beginning we are given an empty array B of size × . Let ussee how to fill in the first row. We have A , = ց , so this means that the first row in B must begin withthe group of ( d − k + ℓk ) − = 3 = 9 integers from 19 to 27 in some order, then continue with the group ofnine integers from 10 to 18 in some order, and finish with the group of nine integers from 1 to 9 in someorder. Then, A , = ր , so each of the three groups above will consist of three smaller groups, ordered inincreasing order. For example, the integers in the first group will be 19, 20, 21 in some order, then wecontinue with 22, 23, 24 in some order, and we finish with 25, 26, 27 in some order. Also, the second groupof nine integers will begin with 10, 11, 12 in some order, continue with 13, 14, 15 in some order, and thenfinish with 16, 17, 18 in some order. Finally, A , = ր , so the members of each of the nine groups of threeconsecutive integers will be ordered in an increasing order. Therefore, the first row of B is given by , , , , , , , , , , , , , , , , , , , , , , , , , , . Let us analyse the second row. Since A , = ր , it will begin with the integers from 1 to 9 in someorder, then it will continue with the integers from 10 to 18 in some order, and will finish with the integersfrom 19 to 27 in some order. Then, A , = ց , so the first group of nine integers will begin with 7,8,9in some order, then it will continue with 4,5,6 in some order, and will finish with 1,2,3 in some order.The two other groups of nine integers are organised similarly, that is, , ,
18; 13 , ,
15; 10 , , for thesecond one and , ,
27; 22 , ,
24; 19 , , for the third one. Finally, A , = ր , so each group of threeconsecutive integers is ordered in an increasing order. The second row of B is thus given by , , , , , , , , , , , , , , , , , , , , , , , , , , . The third row of B is filled in in a similar way: , , , , , , , , , , , , , , , , , , , , , , , , , , . We prove that the array B does not have property ( k, ℓ, d ) . Proof of Theorem 2.2.
We argue by contradiction. Suppose that there exists a subarray D of size k × d ,whose columns form an increasing sequence of k − dimensional vectors (the case of a subarray D of size ℓ × d , whose columns form a decreasing sequence of ℓ − dimensional vectors is analogous). Let c , c , . . . , c d be the indices of the columns of B , which contain the entries of D . Sublemma 2.4.
There exist two integers i , i ∈ [ (cid:0) k + ℓk (cid:1) ] , for which, for both i = i , i , there is a group ofsize ( d − i , which contains at least two groups of size ( d − i − , all of which contain at least one of thecolumns with indices c , c , . . . , c d . roof. Let i be the largest i between and (cid:0) k + ℓk (cid:1) , for which the set c , c , . . . , c d has elements in at leasttwo different groups of size ( d − i − . Note that i is at least two since a group of size d − cannotcontain all d indices c , . . . , c d . Moreover, by the pigeonhole principle, there is a group of size ( d − i − ,which contains at least two indices among c , . . . , c d . For these two indices, there exists an integer i < i ,for which both of them are contained in a group of size ( d − i , but at the same time they are containedin different groups of size ( d − i − (note that the smallest groups are of size one). This proves thesublemma.Now, let i and i be the integers given by Sublemma 2.4. Then, one must have that in each of the k rows of B , which contain entries of D , the order of the groups of size ( d − i − within any group ofsize ( d − i must be increasing, and the order of the groups of size ( d − i − within any group of size ( d − i must also be increasing. This means that in the array A must contain a subarray of size k × ,full of ր . However, this is not possible since the columns of the array A correspond to walks from (0 , to ( k + ℓ √ , k − ℓ √ ) with k steps of the type ր and ℓ steps of the type ց , and the positions of the steps of thetype ր determine a unique such walk. This is a contradiction, which proves that the array B has property ( k, ℓ, d ) . A natural question to ask in light of our results up to now is: for t ≥ , which is the largest integer N = N ( t ) , for which there exists a numerical array of size ( k + ℓ + t ) × N , which does not have property ( k, ℓ, d ) ? The idea from the proof of Theorem 2.2 gives a clue for a lower bound. One could try to find amaximal family of paths with steps within {ր , ց} k + ℓ + t , for which, at any k positions among the k + ℓ + t ,at most one path of the family has only steps of the type ր , and at any ℓ positions among the k + ℓ + t , atmost one path of the family has only steps of the type ց . The question might be reformulated as follows: Question 3.1.
What is the largest possible N = N ( k, ℓ, t ) ∈ N , for which there exists a family of N binaryvectors of length k + ℓ + t such that no two vectors in the family share k entries equal to 1 ( ր ), and notwo vectors in the family share ℓ entries equal to 0 ( ց )? Arguing as in the proof of Theorem 2.2, we conclude the following fact:
Fact 3.2.
For every k, ℓ, t ≥ , d ≥ , there is an array of size ( k + ℓ + t ) × ( d − N ( k,ℓ,t ) , which does nothave property ( k, ℓ, d ) . Sadly, we cannot compute N ( k, ℓ, t ) in general. In the case when the number of − s (or ր − s) isfixed for all the vectors in the family considered in the question above (say, to k + t with t between0 and t , and let t = t − t ), the problem comes down to looking for a family of paths between (0 , and ( k + ℓ + t √ , k + t − ℓ − t √ ) using only steps in the set {ր , ց} , just like in the proof of Theorem 2.2. In thelanguage of coding theory, we are looking for the largest constant-weight code of length k + ℓ + t , distance t , t ) and weight k + t . It is a famous and vastly studied problem in coding theory tocompute this number, often denoted by A ( k + ℓ + t, t , t ) , k + t ) , see [1, 8, 9] for a number oflower and upper bounds on the function A ( · , · , · ) (not to be confused with the array A constructed above).Now, let M ( k, ℓ, t ) = max ≤ t ≤ t A ( k + ℓ + t, t , t ) , k + t ) . Since A ( k + ℓ + t, t , t ) , k + t ) ≤ N ( k, ℓ, t ) for every choice of t between 0 and t , weconclude that M ( k, ℓ, t ) ≤ N ( k, ℓ, t ) as well. As a consequence we obtain following corollary. We believethat it may be used in practice in the construction of large numerical arrays satisfying property ( k, ℓ, d ) for some small values of the parameter: Corollary 3.3.
For every k, ℓ, t ≥ , the construction from the proof of Theorem 2.2 yields an array ofsize ( k + ℓ + t ) × ( d − M ( k,ℓ,t ) , which does not have property ( k, ℓ, d ) . k + ℓ − rows are present. The author would like to thank Emil Kolev for useful remarks on the discussion section, Violeta Naydenovafor a meticulous proofreading, and Peter Boyvalenkov and Ivailo Hartarsky for important corrections.
References [1] A. E. Brouwer. Bounds for binary constant weight codes. .[2] M. Bucić, B. Sudakov, and T. Tran. Erdős-Szekeres theorem for multidimensional arrays, 2019.[3] H. Burkill and L. Mirsky. Combinatorial problems on the existence of large submatrices I.
DiscreteMathematics , 6(1):15 – 28, 1973.[4] H. Burkill and L. Mirsky. Monotonicity.
Journal of Mathematical Analysis and Applications , 41(2):391–410, 1973.[5] P. Erdős and G. Szekeres. A combinatorial problem in geometry.
Compositio mathematica , 2:463–470,1935.[6] P. C. Fishburn and R. L. Graham. Lexicographic Ramsey theory.
Journal of Combinatorial Theory,Series A , 62(2):280–298, 1993.[7] K. Kalmanson. On a theorem of Erdős and Szekeres.
Journal of Combinatorial Theory, Series A ,15(3):343–346, 1973.[8] F. J. MacWilliams and N. J. A. Sloane.
The theory of error correcting codes , volume 16. Elsevier,1977.[9] D. H. Smith, L. A. Hughes, and S. Perkins. A new table of constant weight codes of length greaterthan 28.