aa r X i v : . [ m a t h . C O ] D ec RAMSEY PROPERTIES OF PRODUCTS OF CHAINS
CSABA BIR ´O AND SIDA WAN
Abstract.
Let the product of t chains of equal size be called a t -dimensionalgrid. The main contribution of this paper is the following theorem: For every t -dimensional grid P there exists a (larger) t -dimensional grid Q such that ifwe color every comparability of Q with one of r colors, then there is a subposet P ′ of Q such that P ∼ = P ′ , and every comparability of P ′ is of the same color.We prove other related theorems for graphs, and a variation of our maintool: the Product Ramsey Theorem. Introduction
The original motivation of this paper was two questions of Neˇsetˇril and Pudl´akfrom 1986. In their paper [10], they introduced the notion of the Boolean dimensionof partially ordered sets, and proved an upper and a lower bound based on thenumber of points. At the end of their note they asked two questions. They didnot voice their thoughts about which way the questions would go; nevertheless werephrase the questions as “statements” for easier discussion.
Statement 1.1.
The Boolean dimension of planar posets is unbounded.
Note that this is the opposite of what many researchers, possibly includingNeˇsetˇril and Pudl´ak, conjectured. E.g. in [2], the authors state that it is clearfrom the presentation of the question in [10] that they believed the answer shouldbe “no”.We will say that a class C of posets has the Ramsey-property, if for all r and P ∈ C , there is a poset Q ∈ C , such that for every r -coloring of the comparabilities of Q , there is a subposet Q ′ of Q that is isomorphic to P such that every comparabilityof Q ′ is of the same color.Neˇsetˇril and R¨odl [11] proved that the class of all posets has the Ramsey property.In this paper we are proving a stronger theorem, and the Neˇsetˇril–R¨odl theoremwill follow as corollary.Neˇsetˇril and Pudl´ak asked a second question in their paper, which we also phraseas a statement. Statement 1.2.
The class of planar posets has the Ramsey property.
Neˇsetˇril and Pudl´ak pointed out that Statement 1.2 implies Statement 1.1,though they did not include the proof in their short article. In this paper wefirst provide a proof of this implication. Then we prove two contrasting results: asimple theorem that planar graphs don’t have the Ramsey property, and a morecomplicated one, that grids do have the Ramsey property. We call the latter resultthe Grid Ramsey Theorem.In the latter proof our main tool is the Product Ramsey Theorem. In the last partof the paper, we prove a variation of the Product Ramsey Theorem, which we call the Diamond Ramsey Theorem, but we were only able to prove it in 2 dimensions.The t -dimensional version of the Diamond Ramsey Theorem can be used to provethe t -dimensional version of the Grid Ramsey Theorem, and we do conjecture thatthe Diamond Ramsey Theorem stands in arbitrary number of dimensions.We note that although some of these definitions and questions may be interestingfor infinite posets, in this paper every poset is finite. In fact, we will omit the wordfinite, and even when we say, e.g. “class of all posets”, we mean “class of all finiteposets”. 2. Definitions
First we define Ramsey property of general classes of sets with relations to provea general lemma what shows that the number of colors (as long as it is at least 2)does not matter.Let X be a set, and r a positive integer. An r -coloring of X is a function c : X → [ r ], where [ r ] = { , . . . , r } . The elements of [ r ] are called colors . Sometimeswe will use the r -coloring for other functions with range of cardinality r . A relation R on X is a subset R ⊆ X × X . If X ′ ⊆ X , we use the usual notation c | X ′ and R | X ′ for the restriction of c and R (respectively) to the subset X ′ .Let C be a class of ordered pairs ( X, R ), where X is a set, and R is a relationon X . We say that C has the Ramsey Property , if for all (
X, R ) ∈ C and for all r positive integer, there exists ( Y, S ) ∈ C such that for every c r -coloring of S , thereexists a subset Y ′ ⊆ Y such that if S ′ = S | Y ′ , then ( Y ′ , S ′ ) ∼ = ( X, R ), and for all a, b ∈ S ′ , the c | S ′ ( a ) = c | S ′ ( b ).Less formally C has the Ramsey property, if for all X ∈ C there is a (larger) Y ∈ C , such that if we r -color the relations of Y , we will find a monochromaticsubrelation X ′ of Y that is isomorphic to X . Monochromatic means that everyrelation of X ′ is assigned the same color. The set (with the relation) Y is calledthe Ramsey set of X .With a slight abuse of notation, we will often write X for the pair X = ( X, R ).After this definition one might think that the classical theorem of Ramsey couldbe rephrased by perhaps saying that the set of (complete) graphs have the RamseyProperty, using the usual definition of a graph as an irreflexive, symmetric relation.This is, however, not the case. Clearly, one can 2-color the relations of a graph bycoloring the two directions of an edge opposite colors for each edge, and then nomonochromatic edge will even be found. So we will define the Ramsey Property ofclasses of graphs in a more restrictive way later: by forcing each edge to be coloredthe same.On the other hand, Ramsey’s Theorem can be stated with this terminology: itis the statement that the class of linear orders has the Ramsey Property. This willbe a special case of our Grid Ramsey Theorem.The following lemma is often useful when one is trying to prove that a class hasthe Ramsey Property.
Lemma 2.1.
Suppose C has the following property: for all X ∈ C there is a Y ∈ C and a positive integer r ≥ , such that if we r -color the relations of Y , we willfind a monochromatic subrelation X ′ of Y that is isomorphic to X . Then C has theRamsey Property.Proof. Let C be a class, X ∈ C , and r a positive integer; we need to show that aRamsey set Y can be found. We will do that by induction on r . If r ≤
2, then by
AMSEY PROPERTIES OF PRODUCTS OF CHAINS 3 the conditions there exists Y and r ≥
2. Since r ≤ r , an r -coloring is a special r -coloring, so the statement follows.Now let r >
2, and assume the statement is true for r −
1. So there exists Y ∈ C ,such that if we r − Y , there is a monochromatic subrelation X ′ of Y that is isomorphic to X .We can use the hypothesis again for Y ∈ C and 2-colors. There exist a proper Z ∈ C . We claim that Z is correct choice for the original set X and r colors.To see this, consider an r -coloring c of the relations of Z . Now recolor Z withonly 2 colors based on the c : if c ( x ) = 1, use the color blue; otherwise use the colorred. We know Z yields a monochromatic Y . If Y is blue, then we notice that X isa subrelation of Y , so we found a monochromatic X . If Y is red, then we revert to c to color the relations of Y with r − X this way. (cid:3) For the balance of this paper we assume the reader is familiar with basic notionsof partially ordered sets and graph theory. We refer the reader to the monographof Trotter [13], and the textbook of Diestel [4].3.
Two questions of Neˇsetˇril and Pudl´ak
Recall that the Iverson bracket is a notation that converts a logical propositionto 0 or 1: [ P ] = 1 if P is true, and [ P ] = 0, if P is false.Let P be a poset, and let ( L , S ) be a pair with L = { L , . . . , L d } , ( d ≥
1) is aset of linear orders of the elements of P , and S is a set of binary (0–1) strings oflength d . For two distinct elements x, y ∈ P , let P i ( x, y ) be the proposition that x < y in L i . We call ( L , S ) a Boolean realizer , if for any two distinct elements x, y ,we have x < y in P if and only if [ P ( x, y )][ P ( x, y )] . . . [ P d ( x, y )] ∈ S . We call thisbinary string the signature of the pair ( x, y ). The number d is the cardinality orsize of the Boolean realizer. The minimum cardinality of a Boolean realizer is theBoolean dimension of P , denoted by dim B ( P ).We note that there are minor variations in the the definition of Boolean realizersin the literature. (See next paragraph for citations.) With our definition, antichainsare of Boolean dimension 1 (one can take S = ∅ ), chains are of Boolean dimension1, and in general, dim B ( P ) ≤ dim( P ), because a Dushnik–Miller realizer P can beeasily converted into a Boolean realizer of the same size by taking S = { . . . } .Boolean dimension and structural properties of posets have seen a increasedinterest in recent years. e.g. in [6], the authors showed that posets with covergraphs of bounded tree-width have bounded Boolean dimesnsion. Further, in [1],the authors compared the Dushnik-Miller dimension, Boolean dimension and localdimension in terms of tree-width of its cover graph, and in [8], the authors studiedthe behavior of Boolean dimension with respect to components and blocks.As mentioned earlier, the following statement appeared without proof in [10].We include a proof for completeness. Proposition 3.1.
Statement 1.2 implies Statement 1.1.Proof.
Assume that Statement 1.2 is true, but the Boolean dimension of planarposets is at most k . Let P be a planar poset whose Dushnik–Miller dimensionis greater than k (such poset is well-known to exist). By Statement 1.2, there isa planar poset Q such that any 2 k -coloring of the comparabilities of Q yields amonochromatic P . CSABA BIR ´O AND SIDA WAN
Let ( L , S ) be a Boolean realizer of size k of Q , and let L = { L , . . . , L k } . Colorthe comparabilities of Q with binary strings of length k as colors: if x < y in Q , letthe color of ( x, y ) be the signature of ( x, y ).Now let P ′ be a subposet of Q such that P ′ ∼ = P and every comparable pair of P ′ is of the same color, say d d . . . d k (where d i is the i th digit of the binary string).Let M i = L i if d i = 1, and let M i = L di (the dual of L i ), if d i = 0. It is routine toverify that { M , . . . , M k } is a realizer of P , contradicting dim( P ) > k . (cid:3) Ramsey property of planar graphs
We say that a class of graphs C has the Ramsey Property , if for all G ∈ C andfor all r positive integer, there exists H ∈ C such that for every r -coloring of E ( H ),there exists an induced subgraph G ′ of H such that G ′ ∼ = G , and every edge of G ′ is of the same color. We use the term “monochromatic” as before, and we call thegraph H the Ramsey graph of G .Ramsey’s Theorem can be restated by saying the class of complete graphs hasthe Ramsey Property. The fact the class of all graphs has the Ramsey Property is amore difficult statement, and it was proven around 1973 independently by Deuber[3], by Erd˝os, Hajnal and P´osa [5], and by R¨odl [12].We note that as for general relations, the analogous lemma is true and can beproven exactly the same way. Lemma 4.1.
Suppose C , a class of graphs, has the following property: for all G ∈ C there is a H ∈ C and a positive integer r ≥ , such that if we r -color the edges of H , we will find a monochromatic induced subgraph G ′ of H that is isomorphic to G . Then C has the Ramsey Property. Motivated by our problem on planar posets, we were curious of the class of planargraphs has the Ramsey Property. This turned out not to be the case.
Proposition 4.2.
The class of planar graphs do not have the Ramsey Property.Proof.
It is well-known that the edge set of a planar graph can be partitioned intothree forests. (This is a consequence of the tree covering theorem of Nash-Williams[9], see also Exercise 4 in Chapter 4 of [4].) Let G be a planar graph that is nota forest. Now suppose that the class of planar graphs has the Ramsey Property.Then there exists a planar Ramsey graph H for G . Decompose H into 3 forests,and color the edges of H based on which forest they are in. Let G ′ ∼ = G be amonochromatic induced subgraph of H . Since the edges of G ′ use a single color, G ′ is a forest, a contradiction. (cid:3) Ramsey property of grids
Terminology.
We will use k to denote the k -element chain, and k t for theposet that is the product of the k -element chain by itself t times. This poset will becalled the k t grid or k t diamond, used interchangeably. The number t is called thedimension of the grid/diamond. This coincides with the Dushnik–Miller dimensionof the poset for n ≥
2, so it will not cause confusion.However, we will use the terms subgrid and subdiamond to mean different things.A subdiamond of n t is a subposet isomorphic to a (typically smaller) diamond (butsame dimension). AMSEY PROPERTIES OF PRODUCTS OF CHAINS 5
To define a subgrid, let the ground set of the poset n t be the set of t -tuples ofnumbers from [ n ]. Let S , S , . . . , S t be nonempty subsets of [ n ]. The subposetinduced by the elements of S × · · · × S t is called a subgrid of n t . Of course, everysubgrid is a subdiamond, but not the other way around.We will use the Product Ramsey Theorem, which can be phrased with our ter-minology as follows. Theorem 5.1 (Product Ramsey Theorem [7]) . For all t , r , m , and l there existsan n such that for all r -coloring of the m t subgrids of n t , there is a monochromatic l t subgrid L . That is, every m t subgrid of L received the same color. Our result is as follows.
Theorem 5.2 (Grid Ramsey Theorem) . For each t positive integer, the class of t -dimensional grids have the Ramsey Property. Before the proof we would like to point out how fundamentally different thesetwo theorems are in flavor. In the Product Ramsey Theorem one colors subgrids of n t , while in the Grid Ramsey Theorem, comparabilities are colored. Proof.
Let t, r be positive integers. Let s be a positive integer, and P be an s t grid.We will show that a Ramsey poset Q exists for P .If s = 1, the theorem is trivial, so we assume s ≥ t , r , m = 2, and l = s t , and we geta number n . We claim that Q = n t is a Ramsey poset for P .To show this, we consider a coloring c : C ( Q ) → [ r ] of the comparabilities of Q ; here C ( Q ) denotes the set { ( a, b ) ∈ Q : a ⊥ b } . We will use this to define an r -coloring of the t subgrids of Q as follows. Let M be a t subgrid, with theleast element a , and the greatest element b . Then we assign the color c ( a, b ) to thissubgrid.By the Product Ramsey Theorem, a monochromatic l t subgrid exists; let thisbe called R . We will show that R has a subposet P ′ isomorphic to P such that onthe set of comparabilities { ( a, b ) ∈ P ′ × P ′ : a ⊥ b } , the coloring c is constant. Thiswill finish the proof.To see this last statement, we will prove the following technical lemma. Lemma 5.3.
Let t ≥ , s ≥ be a positive integers. Let B = { [ a , . . . , a t − ] T ∈ Z t : 0 ≤ a i < s t } , and define a poset Q on B with coordinate-wise ordering of itsvectors: [ a , . . . , a t − ] T ≤ [ b , . . . , b t − ] T if and only if a i ≤ b i for all i = 0 , . . . , t − .Now let v = ss ... s t − , v = ss ... s t − , . . . , v t − = s t − s ... s t − , and let L = ( t − X i =0 α i v i : 0 ≤ α i < s, α i ∈ Z ) . Define a poset P on L based on the coefficients: P α i v i ≤ P β i v i if and only if α i ≤ β i for all i . Then P is isomorphic to Q | L . Furthermore, for all j = 0 , . . . , t − , CSABA BIR ´O AND SIDA WAN
Figure 1.
Illustration of Lemma 5.3 for t = 2, s = 4. The 16 × Q , and the subposet P is shown in red. and for all u , w ∈ L , u = w , the j th coordinate of u is different from the j thcoordinate of v .Proof. Let u = P α i v i , w = P β i v i such that u ≤ w in P . Then α i ≤ β i for all i ,so w − u = t − X i =0 ( α i − β i ) v i has all nonnegative coordinates, implying u ≤ w in Q .Now suppose that u = P α i v i , w = P β i v i , and u ≤ w in Q . That is, for each j , the j th coordinate u is less than or equal to the j th coordinate of w . In details,for all j ,(1) α s j + α s j +1 + · · · + α t − j − s t − + α t − j + α t − j +1 s + · · · + α t − s j − ≤ β s j + β s j +1 + · · · + β t − j − s t − + β t − j + β t − j +1 s + · · · + β t − s j − We can view this inequality as the inequality of two nonnegative integers writtenin base s : α t − j − α t − j − . . . α α α t − . . . α t − j ≤ β t − j − β t − j − . . . β β β t − . . . β t − j . This implies α t − j − ≤ β t − j − . Since this is true for all j = 0 , , . . . , t −
1, we getthat u ≤ w in P .To see the last statement, consider u = P α i v i , w = β i v i , and u = w . If the j th coordinate of u is equal to the j th coordinate of w , then the two sides of (1) AMSEY PROPERTIES OF PRODUCTS OF CHAINS 7 are equal. By the uniqueness of base s representation of nonnegative integers, thisimplies that α i = β i for all i , contradicting u = w . (cid:3) In the previous lemma, the poset Q is a grid, and the poset P is a subdiamondof Q . We will refer to the poset P as the core diamond of Q . Note that not everygrid has a core diamond. The grid size (the size of the chains in the product) mustbe of the form s t , so the grid will have s t elements. The core diamond will be thenisomorphic to s t .We now return to the proof of the theorem. Since R is isomorphic to l t , we mayuse t -tuples of numbers from { , . . . , l − } to denote the elements of R , and in turn,these can be considered vectors in Z t . The coordinatewise ordering of these vectorsis the partial order R . Recall that l = s t , and apply Lemma 5.3 to construct thecore diamond of R , which we will call P ′ .Now clearly P ′ is a subposet of R . It is immediate from the definition of P ′ thatit is isomorphic to s t = P . It remains to be proven that P ′ is monochromatic, moreprecisely, every comparability of P ′ received the same color.Let u , w ∈ P ′ such that u < w . By Lemma 5.3, (1) u < w in Z t , and (2) everycoordinate of u and w are distinct. These exactly mean that u and w are the leastand greatest elements of a subgrid M , with M ∼ = t . Since every t subgrid of R received the same color by the coloring c , and since the coloring of comparabilitiesof Q (and so those of R ) were determined by this color, we conclude, that everycomparability of P ′ is of the same color. (cid:3) We would like to note that Ramsey’s classical theorem is a special case of theGrid Ramsey Theorem when t = 1.The theorem of Neˇsetˇril and R¨odl now follows easily. Corollary 5.4.
The class of all posets has the Ramsey Property.Proof.
Let P be a poset. It is well-know that every poset is a subposet of a largeenough Boolean lattice. The Boolean lattice of dimension d is the grid d .So first find a Boolean lattice B such that P is a subposet of B . Then useTheorem 5.2 to find a grid Q , a Ramsey Poset for B . A monochromatic subposet B clearly contains a monochromatic P , so the theorem follows. (cid:3) Diamond Ramsey Theorem and Conjecture
Our original attempt to prove the Grid Ramsey Theorem was via the follow-ing interesting variation of the Product Ramsey Theorem, which is an interestingconjecture on its own right.
Conjecture 6.1.
For all t , r , m , and l there exists an n such that for all r -coloringof the m t subdiamonds of n t , there is a monochromatic l t subdiamond L . That is,every m t subdiamond of L received the same color. However, we were unable to settle this conjecture in the general case. After find-ing our alternative proof for the Grid Ramsey Theorem, we realized that Lemma 5.3can be use to prove Conjecture 6.1 for t = 2. Theorem 6.2.
For all r , m , and l there exists an n such that for all r -coloring ofthe m subdiamonds of n , there is a monochromatic l subdiamond L . That is,every m subdiamond of L received the same color. CSABA BIR ´O AND SIDA WAN
Figure 2.
For the proof of Theorem 6.2. In this figure, m = 2, l = 3. The 9 × Q ′ , the core diamond P ′ is red. The pointsof the subdiamond D are marked by black dots. The blue grid is S . Proof.
Let M = m , and L = l . By the Product Ramsey Theorem, there exists n positive integer such that for all r -coloring of the M subgrids of n , there isa monochromatic L subgrid. We claim that n satisfies the requirements of ourtheorem.To see this, let c be an r -coloring of the m subdiamonds of n . We will definean r -coloring c on the M subgrids of n . For each Q ∼ = M subgrid, let P be thecore diamond of Q , as defined by Lemma 5.3. Let c ( Q ) = c ( P ).As noted earlier, the n grid has a monochromatic L subgrid under the coloring c ; call this Q ′ . Here, monochromatic means that there exists a color r , such thatfor every M subgrid G of Q ′ , we have c ( G ) = r . Let P ′ be the core diamond of Q ′ (see Figure 2).Clearly, P ′ ∼ = l . It remains to be seen that every m subdiamond of P ′ receivedthe same color under c .Let D be an arbitrary m subdiamond in P ′ . Let S = { Proj ( x ) : x ∈ D } S = { Proj ( x ) : x ∈ D } , where Proj i ( x ) is the i th coordinate of x in n . Let S = S × S . By Lemma 5.3, | S i | = | D | = m = M , so S ∼ = M , a subgrid of Q ′ . Therefore c ( S ) = r . Thefollowing claim implies c ( D ) = r , which will finish the proof. Claim 6.3. D is the core diamond of S .Proof of claim. Recall that the core diamond has the property that no two elementshave the same j th coordinates for j = 0 ,
1. We will argue that S has only one m subdiamond with this property. Since D is an m subdiamond of S that does havethis property, we will conclude that D is the core diamond.To see this, let A be a m subdiamond of S with this property. One can thinkof S as a square grid with M rows and M columns. This way, each element of A is in the intersection of a row and a column, and every column and every row of S contains exactly one element of A . This defines two permutations of the elementsof A . Label the elements of A by a , a , . . . , a M − by letting a i be the element of A in the i th column. Then enumerating the elements in the 0th, 1st,. . . rows definesa permutation a σ (0) , . . . , a σ ( M − .Note that L : a < a < · · · < a M − L : a σ (0) < a σ (1) < · · · < a σ ( M − are linear extensions of D , and { L , L } forms a Dushnik–Miller realizer of A .Next we will show that there is only one realizer of two linear extensions of A . This will ensure that σ is determined (independent of A ), and thereby A isdetermined solely by the properties of having exactly one element in each row andcolumn, and being isomorphic to m .To prove this last statement, we will refer the elements of A as vectors in [ m ] × [ m ].Note the following incomparable pairs in A . I = (cid:26)(cid:18)(cid:20) (cid:21) , (cid:20) m − (cid:21)(cid:19) , (cid:18)(cid:20) (cid:21) , (cid:20) m − (cid:21)(cid:19) , . . . , (cid:18)(cid:20) m − (cid:21) , (cid:20) m − m − (cid:21)(cid:19)(cid:27) I = (cid:26)(cid:18)(cid:20) (cid:21) , (cid:20) m − (cid:21)(cid:19) , (cid:18)(cid:20) (cid:21) , (cid:20) m − (cid:21)(cid:19) , . . . , (cid:18)(cid:20) m − (cid:21) , (cid:20) m − m − (cid:21)(cid:19)(cid:27) Now let( x , y ) = (cid:18)(cid:20) i + 10 (cid:21) , (cid:20) im − (cid:21)(cid:19) ∈ I and ( x , y ) = (cid:18)(cid:20) j + 1 (cid:21) , (cid:20) m − j (cid:21)(cid:19) ∈ I Notice that these two incomparable pairs can not be reversed at the same timein a linear extension: indeed, they form an alternating cycle, because x ≤ y , and x ≤ y . So every pair in I must be reversed in a single linear extension, and thesame is true for I . There is only one linear extension that reverses every pair in I , and there is only one for I . (cid:3)(cid:3) It is worth noting why the techniques used to prove this theorem do not workto prove it for higher dimension. This is because the analogue of Claim 6.3 doesnot hold in higher dimension. It was clear from the beginning that a higher dimen-sional grid does not have a unique minimum realizer the same way a 2-dimensionaldoes; but all further hope of salvaging this technique was lost when we found acounterexample for the analogue of Claim 6.3 in 3 dimensions.7.
Closing thoughts
As every finite poset is a subposet of a d -dimensional Boolean lattice for suffi-ciently large d , and as the d -dimensional Boolean lattice is isomorphic to the d grid, we get the Neˇsetˇril–R¨odl Theorem as a corollary of the Grid Ramsey Theorem. Corollary 7.1.
The class of posets has the Ramsey Property.
Furthermore, since every poset of Dushnik–Miller dimension d can be embeddedinto k d for sufficiently large k , the following corollary is immediate. Corollary 7.2.
The class of posets of dimension at most d has the Ramsey Prop-erty. (Of course the corollary remains true if one replaces “at most” with “exactly”.)Unfortunately none of the tools used here seem to be capable of grasping thecomplexities of planar posets. In fact, we set out on our mission trying to disproveStatement 1.2 (hoping that perhaps a grid can serve as a counterexample) to providefurther evidence for Statement 1.1. We still think the most likely outcome is thatboth statements are false. References [1] Fidel Barrera-Cruz, Thomas Prag, Heather C. Smith, Libby Taylor, and William T.Trotter. Comparing Dushnik-miller dimension, Boolean dimension and local dimension.arXiv:1710.09467, 2019.[2] Bart lomiej Bosek, Jaros law Grytczuk, and William T. Trotter. Local dimension is unboundedfor planar posets. arXiv:1712.06099v3, 2020.[3] Walter A. Deuber. Generalizations of Ramsey’s theorem. In
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Department of Mathematics, University of Louisville, Louisville, KY 40292, USA
Email address : [email protected] Department of Mathematics, University of Louisville, Louisville, KY 40292, USA
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