Strictly monotonic multidimensional sequences and stable sets in pillage games
Let have size . We show that there are distinct points such that for each , the coordinate sequence is strictly increasing, strictly decreasing, or constant, and that this bound on is best possible. This is analogous to the Erdős-Szekeres theorem on monotonic sequences in .
We apply these results to bound the size of a stable set in a pillage game.
We also prove a theorem of independent combinatorial interest. Suppose is a set of points in such that the set of pairs of points not sharing a coordinate is precisely . We show that , and that this bound is best possible.
The main theorem of this paper is Theorem 6, which concerns the existence of strictly monotonic sequences in (for some definition of strictly monotonic). The proof of Theorem 6 also requires Theorem 11, a theorem of independent interest. Section 4 describes an application of our results to stable sets in pillage games (this was the original motivation for Theorem 6). We begin by giving some background.
1.1 Non-strict monotonicity
A theorem of Erdős and Szekeres  tells us that within a sequence of real numbers, we can always find a monotonically increasing subsequence of length or a monotonically decreasing subsequence of length . The bound is best possible, as can be seen by considering the sequence
The original proof of Erdős and Szekeres used geometrical reasoning. One can also deduce it from Dilworth’s theorem (or an immediate corollary of it; see Lemma 13) by considering a partial order where in the partial order if and occurs before in the sequence. Then a chain in this partial order corresponds to an increasing subsequence and an antichain corresponds to a decreasing subsequence. We also give a distinct proof due to Seidenberg  below.
A sequence of points with is monotonic in direction if for each , the th coordinate sequence is (not necessarily strictly) decreasing or increasing according to whether or respectively.
We will sometimes omit the direction, so a monotonic sequence in is one that is monotonic in some direction.
A set contains a monotonic sequence of length (in direction ) if there are distinct points such that the sequence is monotonic (in direction ).
There is a rough equivalence between sequences in and sets in . A set in can be ordered by the first coordinate (making an arbitrary choice of ordering when two points share a first coordinate) and projected in the second coordinate to get a sequence in . Conversely, a sequence of real numbers can be mapped to a set in via . These generalize to a rough equivalence between sequences in and sets in . The Erdős-Szekeres theorem thus gives conditions guaranteeing a monotonic sequence in a set in .
If is a monotonic sequence in direction , then is a monotonic sequence in direction , so for sequences in sets, we only need to consider one of or . With this in mind, define
and enumerate this set as .
The generalization of the Erdős-Szekeres theorem to is as follows.
(Non-strict monotonicity.) Let for , and let have size
Then contains a monotonic sequence of length in some direction .
The bound in Proposition 3 is best possible; we give a construction of size containing no such sequence in Section 2.1. The proposition was proved by De Bruijin  in the case for all by using applications of the Erdős-Szekeres theorem. For non-constant , Proposition 3 can be proved by a counting argument of Seidenberg , which we give below.
Let , . Order the points by the first coordinate (so we consider as a sequence of points in ). Assign each position , a -tuple of numbers , where is the maximum length of a subsequence in in direction ending at . Then no tuple of numbers is repeated: given sequence positions , the point must lie in some direction from the point . Then the sequence in direction of length ending at can be extended to a sequence containing , and thus . If no sequence has length then for all , and so by distinctness of the tuples, we have . ∎
1.2 Strict monotonicity
Suppose we wish to find a strictly increasing, strictly decreasing or constant subsequence in a sequence in (we must allow constant subsequences). The Seidenberg counting argument shows that a sequence in with no such subsequence of length has maximum length . This is best possible; consider the example (1) with each replaced by consecutive copies of .
A sequence of points with is strictly monotonic in direction if for each , the th coordinate sequence is strictly decreasing, constant, or strictly increasing according to whether , or respectively.
As before, we will sometimes omit the direction when talking about strictly monotonic sequences.
A set contains a strictly monotonic sequence of length (in direction ) if there are distinct points such that the sequence is strictly monotonic (in direction ).
We need consider only one of each or for each , so define
Consider a set . If we order the points by a coordinate to get a sequence in , then a strictly monotonic subsequence of does not necessarily correspond to a strictly monotonic sequence in (it now matters what happens to points sharing the coordinate that we order by). Thus one cannot apply the counting argument of Seidenberg to bound the size of a set with no strictly monotonic sequence of length . Further, even if we start with a sequence in , the counting argument only gives a bound of , which is far from best possible. Thus we need new techniques to work with strict monotonicity.
1.3 Strict monotonicity in sets
Suppose we wish to construct a large set in with no strictly monotonic sequence of length . Call such a set a good set. Here we describe a natural construction which is in fact largest possible (see Section 2.2 for the exact construction). As mentioned for Proposition 3, there is a set of size with no monotonic sequence of length , and so it is certainly good. In fact the construction in Section 2.1 for contains no pair of points that share a coordinate for any coordinate position. Suppose is another good set such that every pair of points in share a coordinate in some coordinate position. If we replace each point in with a very small copy of to get a new set ( is the “product” of and ; this is made more precise in Section 2.2), then any strictly monotonic sequence in must either have all the coordinate sequences non-constant (thus taking at most one point from each copy of ), or it must lie strictly inside some fixed copy of . In the first case, it corresponds to some monotonic sequence in , and thus has length at most . In the second case it has length at most since is good. Thus is also good. One candidate for is given by the recursive definition and , where . This recursive construction then gives . Our main theorem shows that this is in fact best possible.
Now let be a collection of maximal lengths with for all . We will show that the maximum size of a set with no strictly monotonic sequence in direction of length for all is essentially the same recursive construction, with suitable choices at each stage to maximize the size of the set produced.
Let the function
give the set of positions of the non-zero coordinates. Note that . Define and via and
For , should be thought of (this will be shown) as the maximum size of a good set when the coordinates of the points can only vary in (for all , for ). Similarly, should be thought of as the maximum size of a good set when the coordinates can only vary in , and no two points share a common coordinate from (for all , if and only if ).
If for all , then , and .
(Strict monotonicity in sets.) Let satisfy for all , and let be as above. Let have size
Then contains a strictly monotonic sequence of length in some direction .
In particular, if then contains a strictly monotonic sequence of length .
1.4 Strict monotonicity in sequences
Theorem 6 bounds how large a set can be without containing a strictly monotonic sequence. We would like an analogue of this theorem for sequences. Unlike the non-strict case, such an analogue is not a triviality.
Let be a sequence of points in . Each of the directions in are now non-equivalent for the purposes of the existence of a subsequence in this direction. Suppose for each , we forbid a subsequence of length in direction . Map the sequence to a set as described in Section 1.1, i.e., . The set of maximum lengths for is now , where
Define as in Section 1.2 for the lengths . We would like to apply Theorem 6, but we cannot do this as stated, since we do not have for all . However, we will show that the proof of Theorem 6 still applies for this special case.
(Strict monotonicity in sequences.) Let satisfy for all , and let be as above. Let be a sequence in of length
Then contains a strictly monotonic subsequence of length in some direction .
In particular, if then contains a strictly monotonic subsequence of length .
2 Lower bound constructions
2.1 Construction for non-strict monotonicity
Let be a collection of maximum lengths for the set of directions . We will construct a set of size with no sequence of length in direction , for all . This shows the value appearing in Proposition 3 is best possible.
For convenience, write
(and ). Define, for a set and a vector , the set translation
Define recursively, for , the following collection of sets in .
For , define
(Properties of )
for each , , does not have a monotonic sequence of length in the direction . Further does not have any non-trivial monotonic sequences in any of the directions .
(i) This is immediate from the construction.
(ii) We proceed by induction on . The statement is true for . Let , say , , where . Then
(iii) We proceed by induction on . The statement is true for . It is sufficient to show that for . Then, and we are done. Indeed, suppose for . Then and . Hence
(iv) We proceed by induction on . The statement is true for . Suppose , with . Then again by (ii) we have that is a sequence in direction . Therefore, if we have a sequence of points inside , then either it must lie entirely inside an for some in direction for some (and thus have length at most by the inductive hypothesis); otherwise it lies in direction and can take at most one point from each . ∎
Properties (iii) and (iv) of Lemma 8 show that we can take the set as our construction.
2.2 Construction for strict monotonicity
Let , be as in Theorem 6. In this section, a sequence will mean a strictly monotonic sequence. The set we construct will have size with no sequence of length in direction for all .
In equation (2), , and so
Let be such that
Construct a new set of lengths via
By the inductive hypothesis with the collection there is a set of size , which contains no sequence of length in direction for all . Let be the embedding and scaling of into ,
The set has no sequences of length in direction for all , and no non-trivial sequences in any direction . Further .
Construct another set of lengths via
Then as in Section 2.1, there is a set of size with no sequence of length in direction for all , and no non-trivial sequence in any direction .
Lemma 9 shows that the following construction works.
(Properties of )
has no strictly monotonic sequence of length in direction for all .
(ii) This argument is similar to that for property (iv) in Lemma 8. Let be a strictly monotonic sequence in in direction , where . There are two possibilities for this sequence. If it has a constant coordinate sequence, then the points must lie in some copy of , i.e., for some , and . Then the construction of guarantees that . Otherwise it has no constant coordinate, and so is a sequence in in direction , and so by construction of , . ∎
3 Proof of the upper bound
We begin by giving a theorem of independent interest, required for the proof of the main theorem.
Two points are intersecting if they agree in some coordinate, i.e., for some .
Let be a collection of pairs of points in such that each pair is non-intersecting, but all points are otherwise pairwise intersecting. Then .
This bound can be achieved by taking as pairs the opposing corners in the d-dimensional cube .
Our proof uses exterior algebras, and is reminiscent of a proof of a theorem on intersecting sets given by Alon . We describe them here briefly; for a comprehensive introduction the reader can consult for example Marcus . Given a real -dimensional vector space with basis , the exterior algebra is a dimensional vector space with basis and an associative bilinear operation . For , we identify
The operation is defined to satisfy , and we extend by linearity. In particular, for a set of vectors , the wedge product is non-zero if and only if is an independent set of vectors.
Proof of Theorem 11.
Without loss of generality, the coordinates of the points take values in , i.e., .
We consider the exterior algebra over the real vector space . Label a set of basis elements for as
For each , let be the subspace of spanned by the vectors , and let
be a set of vectors in general position, i.e., any 2 of them are linearly independent. For , let be the vector
Then for , if and only if and intersect. Hence,
We now show that the vectors are linearly independent. Suppose for some constants we have that
For given , the wedge product of the left hand side of this expression with is . Since , we must have . This is true for all and . This shows linear independence.
The vectors lie in the vector space spanned by the vectors of the form , where . Thus by linear independence, we have , i.e., . ∎
Let be integers, and let be intersecting (i.e., for all there exists such that ). Then .
Consider as the finite vector space . Partition into the sets
and all translates. There are such sets, and any intersecting subset of takes at most one point from each set. ∎
Let be a partially ordered finite set, with maximum chain length . Then contains an antichain of size at least .
The set of maximal elements in is an antichain, and the set has maximal chain length . Proceeding by induction on with the set gives a partition of into antichains, one of which has size at least . ∎
3.1 Intersecting flats
We consider axis-aligned affine subspaces in , which we will refer to as flats. Label such a flat via a string of numbers and s, say, , where is a wildcard. Thus a point lies in if for all , either or . For example, represents a line in parallel to the -axis through the point . Observe that the dimension of the flat is the number of coordinates.
A set of flats is intersecting if for all we have for some .
Equivalently, a set of flats is intersecting if any pair of points taken from a single flat or a pair of flats in intersect.
An intersecting set of flats is minimal if no flats in can be enlarged (by replacing a coordinate with a ) while remains intersecting.
Equivalently, is intersecting and minimal if is intersecting and further for all and all with , there is some such that and for , either or .
Given a set of pairwise intersecting points , we can construct an intersecting and minimal set of flats containing all the points of as follows. Initially, let , considering as an intersecting set of flats (each of dimension ). Either is minimal, or we can enlarge one of the flats by replacing a coordinate with a , taking only one copy if this produces a duplicate flat. Continue in this manner until no further enlargements can be made.
The set of 3 flats listed below is an example of such an intersecting and minimal system in .
We can use Theorem 11 to bound the number of non- values appearing in each coordinate position for a minimal set of intersecting flats.
Let be an intersecting and minimal set of flats in . Then for all ,
Without loss of generality, we will bound the number of values occuring in the first coordinate. We may assume that the values occuring here are . For each , there are such that , but either or for . Project the last coordinates of to form respectively, where we replace s in with 1, and s in with 2. (There is nothing special about these two values other than that they are distinct.) Then and do not intersect. However, for , must intersect since is intersecting and . The collection of pairs satisfies the conditions in Theorem 11 and the result follows. ∎
3.2 Proofs of the main theorems
Proof of Theorem 6.
We proceed by induction on . The result is clearly true for , so we may assume . Let , be as in Theorem 6. Let be a set of points containing no strictly monotonic sequence of length in direction , for all . We aim to show that .
Each direction corresponds to a partial order on , where we say if is a strictly monotonic sequence in direction . The maximum chain length in the order corresponding to is . Recursively construct a sequence of sets as follows. Let . For , partially order with . Lemma 13 guarantees an antichain of size . Set . No two points in form a sequence in any of the directions , and so every pair of points agree in some coordinate ( is pairwise intersecting). We have
It remains to show that
(for some ).
Define a function
to give the coordinate positions of the -coordinates of a flat, i.e., if and only if .
Our inductive hypothesis tells us the following. Let be a flat and let be the set of -coordinates of . Let be the points of that lie in the flat . Then . Indeed, let and let project in the coordinate positions . For , let be the direction such that and for . Define the set of lengths via
This gives a collection with . Then by the inductive hypothesis with this collection of lengths, .
In particular, let be an intersecting and minimal set of flats that contain all the points of . Then
It is sufficient to show that the right hand side of (5) is at most . If all the points of lie in a flat of dimension , i.e., all points share a single fixed coordinate position , then and we are done by the maximality of . So we will assume from now on that there is no single fixed coordinate, and all the flats have dimension at most .
Let be a set of coordinate positions. Take a chain of subsets with . Since for all , we have that and hence
Thus ( is the value of when for all ). Thus from (5), it is sufficient to prove the inequality
We consider the specific cases , before proving (7) for general . The cases of small are necessary to consider as we are inducting on .
By assumption consists only of flats of dimension , i.e., points. By Lemma 16, at most one value appears in each coordinate. However is minimal (Definition 15), so there are at least two values appearing in each coordinate. This contradiction implies that cannot exist (the only valid is one containing a single 1-dimensional flat).
We aim to show that . By Lemma 16, there are at most 2 non- values used in each coordinate in , say, . If all flats have dimension 0, then and we are done.
Otherwise, say, contains the flat . By minimality, must contain a flat of the form and a flat of the form , with , and no other forms are possible. Since is intersecting, . Further by minimality, . Thus .
We aim to show that . By Lemma 16, there are at most 4 non- values used in each coordinate in , say, .
Suppose first that there are only flats of dimension 0 or 1. Construct a bipartite graph with vertex classes as follows. Let be the set of all flats with exactly one coordinate. Let . Add an edge between and if and differ only in the unique -coordinate of . is thus a 4-regular bipartite graph. The set of flats corresponds to an independent set of vertices in . By Hall’s Theorem, has a matching of size , whose set of end vertices in is an intersecting subset of . This has maximum size by Lemma 12, hence and as required.
So suppose there is a flat of dimension 2, say,