Monochromatic Boxes in Colored Grids

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1 Monochromatic Boxes in Colored Grids Joshua Cooper, Stephen Fenner, and Semmy Purewal October 16, 008 Abstract A d-dimensional grid is a set of the form R = [a 1] [a d ] A d- dimensional box is a set of the form {b 1, c 1 } {b d, c d } When a grid is c-colored, must it admit a monochromatic box? If so, we say that R is c-guaranteed This question is a relaxation of one attack on bounding the van der Waerden numbers, and also arises as a natural hypergraph Ramsey problem viz the Ramsey numbers of hyperoctahedra We give conditions on the a i for R to be c-guaranteed that are asymptotically tight, and analyze the set of minimally c-guaranteed grids 1 Introduction A d-dimensional grid is a set R = [a 1 ] [a d ], where [t] = {1,, t} For ease of notation, we write [a 1,, a d ] for [a 1 ] [a d ] The volume of R is d a i A d-dimensional box is a set of d points of the form {x 1 + ɛ 1 s 1,, x d + ɛ d s d ɛ i {0, 1} for 1 i d}, with s i 0 for all 1 i d A grid R is c, t-guaranteed, if for all colorings f : R [c], there are at least t distinct monochromatic boxes in R, ie, boxes B j R, j [t], so that fb j = 1 When t = 1, we simply say that R is c-guaranteed If R is not c-guaranteed, we say it is c-colorable Clearly, whether a grid is c, t-guaranteed depends only on a 1,, a d Furthermore, if b i a i for all i such that 1 i d, then [b 1,, b d ] is c-guaranteed if [a 1,, a d ] is This ordering on d-tuples is sometimes called the dominance order, and we will denote it by Then one may state the above observation as the fact that the set of c-guaranteed grids is an up-set in the N d, -poset Hence, we have a full understanding of this family if we know the minimal c-guaranteed grids, an antichain in the order Note that any such antichain is finite, a wellknown fact in poset theory Call the set of minimal c-guaranteed grids Oc, d, the obstruction set for c colors in dimension d We will focus our attention on monotone obstruction set elements, ie, those grids for which a 1 a d, Partially supported by NSF grant CCF

2 since being c-guaranteed or c, t-guaranteed is invariant under permutations of the a j The subject of unavoidable configurations in grids has connections with the celebrated Van der Waerden s and Szemerédi s Theorems See, for example, [], [5], and [8] Our results can be seen as belonging to hypergraph Ramsey theory, as follows Let G be the complete d-partite d-uniform hypergraph with blocks of size a 1,, a d Then an edge of G can be identified with a vertex of R = [a 1,, a d ] in the natural way Under this correspondence, a c-coloring of R gives rise to a c-edge coloring of G, and boxes correspond precisely to subgraphs isomorphic to the generalized octahedron K d, the complete d-partite d- uniform hypergraph with each block of size The generalized octahedra play an important and closely related role in the work of Kohayakawa, Rödl, and Skokan [7] on hypergraph quasirandomness Among other interesting results, they show that, asymptotically, a random c-edge coloring of G has the fewest number of monochromatic K d s possible We may translate each of our results into statements about the Ramsey numbers of hyperoctahedra-free d- partite d-uniform graphs For example, in Section 7, we give a family of upper bounds on the sizes of 3-dimensional grids which have a -coloring admitting no monochromatic box; this is equivalent to asking for the extremal tripartite 3-uniform hypergraphs which are K 3, K 3 -Ramsey The present work is even more closely connected to the Product Ramsey Theorem Though the proof appears in [6], the statement appearing in [9] best illustrates the connection: Theorem 11 Product Ramsey Theorem Let k 1,, k d be nonnegative integers; let c and d be positive integers; and let m 1,, m d be integers with m i k i for i [d] Then there exists an integer R = Rc, d; k 1,, k d ; m 1,, m d so that if X 1,, X d are sets and X i R for i [d], then for every function f : X 1 k 1 Xd k d [c], there exists an element α [c] and subsets Y 1,, Y d of X 1,, X d, respectively, so that Y i m i for i [d] and f maps every element of X 1 k 1 Xd k d to α This result ensures that the quantity Nc, d = Rc, d; 1 d ; d, which corresponds to the least R so that [R] d is c-guaranteed, is finite A closer analysis of Nc, d in fact, the more general Nc, d, m = Rc, d; 1 d ; m d appears in the manuscript [1] by Agnarsson, Doerr, and Schoen They obtain asymptotic bounds on Nc, d, m that are valid for large m Here, we examine instead the least nontrivial case of m =, and consider grids which are not necessarily equilateral In the next section, we show that any grid of sufficiently small volume approximately c d 1 is c-colorable The following section shows that the analysis is tight: there are grids of this volume which are c-guaranteed Not all grids of sufficient volume are c-guaranteed, although Section 4 demonstrates that any grid all of whose lower-dimensional subgrids are sufficiently voluminous is indeed c-guaranteed The next section gives a tight upper bound on the volume of minimally c-guaranteed grids, ie, elements of the obstruction set Section

3 6 then addresses the question of how many obstructions there are Finally, as mentioned above, Section 7 considers the case of c = and d = 3, where some interesting computational questions arise This extends work of the second two authors [4] for d = and c 4 Throughout the present manuscript, unless we explicitly say otherwise, we use the notations x = Oy and y = Ωx to mean that there is a function F : Z + Z + such that x F dy That is, x is bounded by y times a number that only depends on d Naturally, x = Θy means that x = Oy and x = Ωy, and notation x = oy is defined analogously In general, x and y will depend on c, d, and perhaps other quantities All small grids are c-colorable Define V c, d to be the largest integer V so that every d-dimensional grid R with volume at most V is c-colorable Below, we show that V c, d is Θc d 1 Theorem 1 V c, d = Ωc d 1 In fact, V c, d > c d 1 /e d, where e = 718 is Napier s constant Proof We apply the Lovász Local Lemma see, eg, [3], which states the following Suppose that A 1,, A t are events in some probability space, each of probability at most p Let G be a dependency graph with vertex set {A i } t, ie, a graph so that, whenever a set S of vertices induces no edges in G, then S is a mutually independent family of events Then P t A i > 0 if ep +1 1, where = G is the maximum degree of G Now, suppose R = [a 1,, a d ] is a grid of volume V, and we color the points of R uniformly at random from [c] Enumerate all boxes in R as B 1,, B t Define A i to be the event that B t is monochromatic in this random coloring Clearly, we may take G to have an edge between A i and A j whenever B i B j The degree of a vertex A i is then the number of boxes B j, j i, which intersect B i Since we may specify the list of all such boxes by choosing one of the d points of B i, and then choosing the d coordinates of its antipodal point, deg G A i is at most d d d a i 1 1 < d a i 1 = d V 1 The outermost 1 here reflects the fact that B i may be excluded among these choices The probability of each A i is the same: p = c d +1 Therefore, which is 1 whenever V c d 1 /e d ep + 1 < ec d +1 d V 3

4 3 Some large grids are c-guaranteed Theorem 31 Fix c, d, define R = [a 1,, a d ], and let M = i ai denote the total number of boxes in R For min{a 1,, a d }, R is c, M1 + o1/c d 1 -guaranteed Theorem 31 follows quickly from the next lemma, whose extra strength we will need later Lemma 3 Suppose c 1 For d 1 and integers a 1,, a d, let M = d ai The grid R = [a1,, a d ] is c, M d /c d 1 -guaranteed provided 1,, d > 0, where j, 0 j d, is given by the recurrence 0 = 1, j = j 1 1 c j 1 j 1 1 a j 1 Proof We proceed inductively Suppose d = 1, let f : [a 1 ] [c] be a c-coloring, and define γ i = f 1 i to be the number of points colored i, 1 i c Then the number N of monochromatic boxes in f is exactly c γi N = = 1 c c γ 1 i γ i = γi a 1 Applying Cauchy-Schwarz, N c γ i c = a 1a 1 c c a 1 = a 1 c a 1 a1 = 1 c a1 c a 1 1 = 1 c a1 1 Now, suppose the statement is true for dimensions < d + 1, and consider a coloring f : [a 1,, a d+1 ] [c] Consider the a d+1 colorings f j of the d- dimensional grid [a 1,, a d ] induced by setting the last coordinate to j, ie, f j x 1,, x d = fx 1,, x d, j Let γ i B, for a box B [a 1,, a d ] and i [c], denote the number of j so that f j B i Then the number N of monochromatic d + 1-dimensional boxes in f is N = γi B i B 4

5 = 1 γ i B γ i B i B B i γ ib 1 Mc γ i B B i = B i γ ib Mc B i γ ib Mc where M = d ai Since, by the inductive hypothesis, fj induces at least M d /c d 1 monochromatic boxes, so that γ i B a d+1m d, c d 1 i B N a d+1 M d /c d+1 a d+1 M d c/c d 1 Mc = a d+1ma d+1 d cd d = M = = c d+1 1 d+1 c d+1 1 d+1 c d+1 1 ad+1 ai d ai d+1 c d+1 1 ad+1 d cd d a d+1 1 a d+1 c d / d a d+1 1 Proof of Theorem 31 Fix c, d 1 It is clear by induction on j that for all 1 j d, as min{a 1,, a d }, j = 1+o1, and so in particular, j > 0 if min{a 1,, a d } is large enough Note that, in the notation of Lemma 3, if 1,, d > 0, then [a 1,, a d ] is not c-colorable Therefore we may conclude the following Corollary 33 In the notation of Lemma 3, let Γ j, 0 j d, be given by the recurrence Γ 0 = 1, Γ j = Γ j 1 1 cj 1 /Γ j 1 a j 1 If Γ 1,, Γ d > 0, then [a 1,, a d ] is c-guaranteed = Γ j 1 Γ j 1 cj 1 a j 1 5

6 Proof Assume Γ 1,, Γ d > 0 A routine induction shows that Γ j j for 0 j d Lemma 34 In the notation of Lemma 3, let ε j be given by the recurrence ε 0 = 0, ε j = ε j 1 + cj 1 If ε d < 1, then [a 1,, a d ] is c-guaranteed a j 1 Proof Clearly, 0 = ε 0 < ε 1 < ε < < ε d, and so by assumption ε i < 1 for all i [d] An induction on i shows that Γ i 1 ε i for 0 i d: This is clearly true for i = 0 Suppose i < d and Γ i 1 ε i Then setting η := c i /a i+1 1 and noting that Γ i 0, we have The term in the parentheses is positive: Γ i+1 = Γ i Γ i η Γ i 1 ε i η 1 1 ε i η 1 ε i η = 1 ε i+1 > 0 by assumption Thus continuing 1 and using the inductive hypothesis again, Γ i 1 ε i η 1 ε i 1 ε i η 1 ε i η = 1 ε i+1 Motivated by the preceding lemma, for every c, d 1 and grid R = [a 1,, a d ] with a i for all i [d], we define ε c R := d ci 1 d i a i 1 Lemma 35 If R = [a 1,, a d ] is not c-guaranteed, then ε c R 1 Proof We let ε j := ε c [a 1,, a j ] = j j i c i 1 /a i 1 for all j with 0 j d, and notice that the ε j satisfy the recurrence in Lemma 34 Corollary 36 For any fixed d 1 and c, if n is least such that [n] d is c-guaranteed, then n < d + c d 1 Furthermore, d e 1 < V c, d c d 1 < d + d dd 1/ Proof If we take a j = d + 1 d j c j for all 1 j d, then ε c R = d/d + 1 < 1 The second result now follows from the fact that d a j < j=1 d d + d j c j 1 j=1 6

7 = d + d d j=1 d j c d j=1 j 1 = d + d d 1 j=1 j c d 1 j=0 j = d + d dd 1/ c d 1 The first result follows by taking n := a d 4 Hereditarily large grids are c-guaranteed It is possible for grids of arbitrarily large volume to be c-colorable Indeed, one need only have one of the dimensions be at most c, and then color the grid with this coordinate However, if we require that each lower dimensional sub-grid be sufficiently voluminous, then the whole grid is c-colorable This statement is made precise by the following theorem Theorem 41 Fix d > 0, and define C j = d d 3 3j 1 1 for j 1 For all integers c 1 and 1 a 1 a a d, if j a i > C j c 3j 1/ for all j [d], then [a 1,, a d ] is c-guaranteed We require a lemma and a bit of notation: If R = [a 1,, a d ] and 1 j < d, let R j denote [a 1,, a j ] and let R j denote [a j+1,, a d ] Note that, if R is c- guaranteed, then R j is as well Indeed, if f : R j [c] is a c-coloring of R j, then the function g : R [c] defined by gx 1,, x d = fx 1,, x j is a c-coloring of R We will also make repeated use of the following easily verified fact: For every integer j 0, j j 1 3 j 1/ and j j j Lemma 4 Let c 1, let R = [a 1,, a d ] be a grid, and let j [d 1] Define j c ai j := c j c a i If R j is c-guaranteed and R j is c -guaranteed, then R is c-guaranteed Proof Assume that R j is c-guaranteed and that R j is c -guaranteed Suppose that f : R [c] is a c-coloring Consider the coloring g : R j [c ] that assigns the pair B, s to the point v, B being an arbitrary choice of j-dimensional box colored monochromatically by f j : R j [c], where f j x 1,, x j = fx 1,, x j, v, and s being its color Note that R j is c-guaranteed, so such a B always exists Then g is a c -coloring, because there are exactly c many different B, s Since R j is c -guaranteed, g colors some d j-dimensional box B 1 monochromatically, with color B, s But then B B 1 is a d-dimensional box monocolored by f with color s Proof of Theorem 41 The statement is clearly true when d = 1 since C 1 = 1 Suppose d > 1 and the statement is true for all d < d Let R = [a 1,, a d ] be a monotone grid satisfying the hypothesis of the theorem Case 1: ε c R < 1 The result follows immediately from Lemma 35 7

8 Case : ε c R 1 Then there is some j [d] such that d j c j 1 /a j 1 1/d, ie, a j d d j c j < d d j+1 c j 1 Since j j 3 j 1 for all integers j 1, j a i j a j < d j jd j+1 c jj 1 d j jd j+1 c 3j 1/, and so for all k [d j], k a j+i > C j+k j+k 1/ 3 j d j c3 1/ jd j+1 C j+k j 3 k d j c3 1/ jd j+1 Let c = d j jd j+1 c 3j Note that c c j a i Then for all k [d j], k a j+i > C j+k d j jd j+1 c d j jd j+1 = dd 3 3j+k 1 1 c 3 k 1/ d d j+1 j3k d d 3 3j+k 1 1 j3 k c 3 k 1/ 3 k 1/ d d 3 3j+k j 13 k / c 3 k 1/, because j 3 j 1/ for all j 1 Continuing the computation, k a j+i > d d 3 3j+k j 13 k / c 3 k 1/ = d d 3 3j+k j+k 1 +3 k 1 c 3 k 1/ = d d 3 3k 1 1 c 3 k 1/ = C k c 3 k 1/ Therefore R j = [a j+1,, a d ] is c -guaranteed by the inductive hypothesis It is easy to see that the C j s are increasing in d, so taking d = d j causes no problem here Since R j is also c-guaranteed by the inductive hypothesis, we may apply Lemma 4 to conclude that R is c-guaranteed 5 Upper bounds on the volume of obstruction grids Before proceeding, we introduce the following notation For d 1 and any monotone grid R = [a 1,, a d ] where a d > 1, we let R denote the monotone 8

9 grid obtained from R by subtracting one from a j, where j [d] is least such that a j = a d Note that if R is monotone and R Oc, d, then R is c-guaranteed but R is not c-guaranteed The next theorem gives an asymptotic upper bound on the volume d a i of any grid [a 1,, a d ] Oc, d Theorem 51 For every d 1 and every grid R = [a 1,, a d ] Oc, d, d a i = O c 3d 1/ The theorem follows immediately from the following lemma: Lemma 5 For every d 1, every c, and every monotone grid R = [a 1,, a d ] Oc, d, there is a set P [d] such that 1 d P, l a i = O c 3l 1/ for every l P, and 3 For every k [d], a k = O c 3j l j 1, where l is the least element of P that is k, and j is the biggest element of P that is < k j = 0 if there is no such element We call the elements of P pinch points for R Proof Let d 1 and c be given, and let R = [a 1,, a d ] Oc, d be a monotone grid Then R is c-guaranteed, and thus R j is also c-guaranteed for all 1 j d Since R Oc, d, we have that R is not c-guaranteed, and thus ε c R 1 This in turn implies that there is some largest l [d] such that cl 1 d l a l 1 d Note that the denominator is positive, because a l a 1 c since R is c-guaranteed Thus, and thus a l d d l c l 1 + d + d l c l 1, l a i a l l d + d l l c l l 1 d + d 1 d c 3 l 1/, 3 which implies that l satisfies Condition of the lemma We will make l the least element of P, noticing that Equation and the monotonicity of R imply that a k satisfies Condition 3 of the lemma for all k [l] with j = 0 9

10 If l = d, then we let P = {l} = {d} and we are done Otherwise, l < d Note that R = R l R l up to a possible permutation of the coordinates Recall also that R l is c-guaranteed, but R is not It follows from Lemma 4 that R l is not c -guaranteed, where c := c l ai = O c The bound in Equation 3 gives c = O c 3l l a i We thus have ε c R l 1, and so there is some largest m with l < m d such that which gives For the volume of R m, we get m a i = d m c m l 1 a m 1 d l, a m d l d m c m l d l + d m c m l 1 5 = O c 3l m l 1 6 l a i m i=l+1 a i l a i a m m l = O c 3l 1/ O = O c 3l 1/ c 3l 3 m l 1/ = O c 3m 1/ 3l m l m l 1 c We make l and m the two least elements of P, and the last calculation shows that m P satisfies Condition Further, since a k a m for all k such that l < k m, Condition 3 is also satified for all these a k by Equations 4 6 If m = d, then we let P = {l, m} and we are done Otherwise, we repeat the argument above using m instead of l to obtain an n with m < n d such that l, m, and n being the least three elements of P satisfies Conditions and 3 of the lemma, and so on until we arrive at d, whence we set P := {l, m, n,, d} The next proposition shows that the bounds in Lemma 5 are asymptotically tight 10

11 Proposition 53 For c, there is an infinite sequence {µ j c} j=1 of positive integers such that 1 µ j c j 1 / c 3j 1 for all j Z +, and for all d 1, the grid [µ 1 c,, µ d c] Oc, d with pinch point set P = [d] Proof For all c, define µ 1 c := 1 + c, c + 1 µ c := 1 + c, µ j+1 c := 1 + c j µi c, Fix c and let µ j denote µ j c for short A routine induction on j shows 1 For the inductive step, noting that j 1 i=0 3i = 3 j 1/, we have µ j+1 = 1 + c 1 + c j 1 + c j = 1 + j µi j µ i 1 j c 3i 1 3i 1 1 c 3j 3j 1/ For, we use induction on d 1 to show separately that 1 [µ 1,, µ d ] is c-guaranteed, and [µ 1,, µ d ] is not c, -guaranteed ie, there is a coloring [µ 1,, µ d ] [c] that monocolors exactly one box Clearly [µ 1 ] = [1+c] is c-guaranteed by the Pigeonhole Principle Now let d and assume that [µ 1,, µ d 1 ] is c-guaranteed Then letting c = c d 1 µi, we have µ d = 1 + c, and hence [µ d ] is c -guaranteed But then, [µ 1,, µ d ] is c-guaranteed by Lemma 4 letting j = d 1 Now for claim For d = 1, clearly the coloring [µ 1 ] [c] mapping j j mod c + 1 has exactly one monochromatic 1-dimensional box, namely, 11

12 1; c = {1, c + 1} Now let d and assume claim holds for d 1, ie, there is a coloring [µ 1,, µ d 1 ] [c] that monocolors exactly one box We will call such a coloring minimal This generates exactly d 1 µi many boxes in [µ 1,, µ d 1 ] For each of these boxes B and for each color s, we can find a minimal coloring that monocolors B with s by permuting the order of the hyperplanes along each axis and by permuting the colors Thus there are exactly c = c d 1 µi many distinct minimal colorings We overlay these c many colorings to obtain a coloring of [µ 1,, µ d 1, c ] with no monochromatic d-boxes We then duplicate the first d 1-dimensional layer to arrive at a c-coloring of [µ 1,, µ d 1, 1 + c ] = [µ 1,, µ d ] This coloring has only one monocolored d-box: the box corresponding to the duplicated layer of unique monocolored d 1-boxes This shows Item It follows from claim that [µ 1,, µ d 1, µ d 1] is not c-guaranteed for any d 1, since we can remove a single hyperplane from the only monocolored d-box in some minimal coloring of [µ 1,, µ d 1, µ d ] to leave a coloring of [µ 1,, µ d 1, µ d 1] without any monochromatic d 1-boxes From this it easily follows that [µ 1,, µ d ] Oc, d, because [µ 1,, µ j 1, µ j 1] is not c-guaranteed, and hence [µ 1, µ j 1, µ j 1, µ j+1,, µ d ] is not c-guaranteed, for any j [d] Finally, it is evident that all j [d] are pinch points for [µ 1,, µ d ] It is interesting to note that [µ 1,, µ d ] is the lexicographically first element of Oc, d 6 Upper bound on the size of the obstruction set It was shown in [4] that Oc, c We give an asymptotic upper bound for Oc, d for every fixed d 3 Theorem 61 For all d 3, Oc, d = O c 17 3d 3 1/ Proof Fix d 3 We give an asymptotic upper bound on the number of monotone grids in Oc, d The size of Oc, d is at most d! times this bound, and so it is asymptotically equivalent By Lemma 5, every grid R Oc, d has a set P of pinch points For each set P [d] such that d P, let # c P be the number of monotone grids in Oc, d having pinch point set P There are d 1 many such P, so an asymptotic bound on max{# c P P [d] d P } gives the same asymptotic bound on Oc, d Fix a set P [d] such that d P, and let P = {l 1 < l < < l s = d}, where s = P and l 1,, l s are the elements of P in increasing order For convenience, set l 0 := 0 Lemma 5 says that for any monotone grid R = [a 1,, a d ] Oc, d having pinch point set P, for any b [s], and for any k 1

13 such that l b 1 < k l b, we have a k = O c eb, where eb := 3 l b 1 l b l b 1 1 To bound # c P, we first note that for any choice of 1 a 1 a d 1, there can be at most one value of a d such that [a 1,, a d ] Oc, d, because any two d-dimensional grids that share the first d 1 dimensions are comparable in the dominance order Thus # c P is bounded by the number of possible combinations of values of a 1,, a d 1 From the bound on each a k above, we therefore have s 1 # c P = O l b b=1 k=l b 1 +1 s 1 b=1 = O c h 1+h O c eb c eb l b l b 1 O where h = esd 1 l s 1 and s 1 h 1 = ebl b l b 1 b=1 d 1 k=l s 1+1 O c es c es d 1 l s 1 s 1 = 3 l b 1 l b l b 1 1 l b l b 1 b=1 s 1 3 l b 1 3l b l b 1 1 b=1 = 1 s 1 3 l b 3 l b 1 b=1 = 3m 1, where m = l s 1 We also have h = 3 ls 1 d ls 1 1 d 1 l s 1 = 3 m d m 1 d m 1, whence h 1 + h = 3m m d m 1 d m 1 So our bound on the exponent of c only depends on the value of m, which satisfies 0 m < d It is more convenient to express h 1 + h in terms of n := d m, where n [d]: h 1 + h = 3d n d n n 1 n 1 13

14 = 3d 1 + n n 1 3 n 1 It is easy to check that 1 + n n 1/3 n is greatest and thus h 1 + h is greatest when n = 3 It follows that which proves the theorem h 1 + h 3d = 17 3d 3 1, The first few values 17 3 d 3 1/ are given in the Figure 1 d 17 3 d 3 1/ Figure 1: Table of upper bounds on e so that Oc, d = Oc e for small d 7 Three Dimensions and Two Colors The following graph Figure, generated using the Jmol module in SAGE and table Figure 3 display upper bounds for the smallest a 3 so that [a 1, a, a 3 ] is -guaranteed All three graphical axes run from 3 to 130; the table includes only 3 a 1 1 and 3 a 1 We believe these values to be very close to the truth; indeed, we have matching lower bounds in many cases, and lower bounds that differ from the upper bounds by at most in many more cases A few different methods were applied to obtain these bounds First, the values j, as in Section 3, were computed, and the least a 3 so that 3 > 0 was recorded In fact, this idea was improved slightly by applying the observation that, if some grid is, t-guaranteed, then it is, t -guaranteed In some cases, this increases the value of j Second, we used the simple observations that c-colorability is independent of the order of the a i, and that R R when R is c-guaranteed implies that R is c-guaranteed Third, we applied the following lemma Lemma 71 If the grid R = [a 1,, a d ] is c, t-guaranteed, then R [ cm/t + 1] is c-guaranteed, where M = d aj j=1 Proof Note that K = cm/t + 1 > cm/t and is integral If we think of R [K] as K copies of R, then any c-coloring of R [K] restricts to K c- colorings of R Since R is c, t-guaranteed, each of these c-colorings gives rise to 14

15 Figure : Graph of upper bounds on a 3 so that [a 1, a, a 3 ] is -guaranteed Figure 3: Table of bounds on a 3 so that [a 1, a, a 3 ] is -guaranteed 15

16 t monochromatic boxes Hence, in K colorings, there are at least t cm/t +1 > cm monochromatic boxes Since there are only M total boxes in each copy of R, and any monochromatic box can only be colored in c different ways, there must be two identical boxes in two different copies of R which are monochromatic and have the same color This is precisely a monochromatic d + 1-dimensional box in R [K] Therefore, in order to obtain upper bounds on [a 3 ] in the above table, we need to know the greatest t for which [a 1 ] [a ] is, t-guaranteed To that end, we define the following matrix: Definition 7 Let M r be the r r integer matrix whose rows and columns are indexed by all maps f j : [r] [], 0 j < r The i, j-entry of M r is defined to be f 1 i 1 f 1 j 1 + f 1 i f 1 j Then define the quadratic form Q r : R r R by Q r v = v M r v Let δ r = M r 1, 1,, M r r, r, the diagonal of M r Proposition 73 Let t be the least value of Q r v v δ r over all nonnegative integer vectors v Z r with v 1 = s Then [r] [s] is c, t-guaranteed, and t is the minimum value so that this is the case Proof Given a vector v = v 1,, v r satisfying the hypotheses, consider the r s matrix A with v j columns of type f j for each j [r] We may identify f j with a column vector in [] r in the natural way It is easy to see that Q r v δ r exactly counts twice the number of monochromatic rectangles in A, thought of as a -coloring of the grid [r] [s] We applied standard quadratic integer programming tools XPress-MP to minimize the appropriate programs Fortunately, for the cases considered, the matrix M r was positive semidefinite, meaning that the solver could use polynomial time convex programming techniques during the interior point search We conjecture that this is always the case Conjecture 74 M r is positive semidefinite for r 3 In particular, for r = 3, the eigenvalues of M r are 0, 1, and 4, with multiplicities, 4, and, respectively For 4 r 9, the eigenvalues are 0, r, r 3 r, r r 1, and r 4 r r +, with multiplicities r rr +1/, rr 1/ 1, r 1, 1, and 1, respectively We conjecture that this description of the spectrum is valid for all r 4 References [1] G Agnarsson, B Doerr, T Schoen, Coloring t-dimensional m-boxes Discrete Math 6 001, no 1-3,

17 [] M Axenovich, J Manske, On monochromatic subsets of a rectangular grid, preprint [3] N Alon, J H Spencer, The probabilistic method Third edition Wiley- Interscience John Wiley & Sons, New York, 008 [4] S Fenner, W Gasarch, C Glover, S Purewal, Rectangle Free Coloring of Grids, preprint [5] H Furstenberg, Y Katznelson, An ergodic Szemerdi theorem for commuting transformations, J Analyse Math , [6] R L Graham, B L Rothschild, J H Spencer, Ramsey theory Second edition Wiley-Interscience Series in Discrete Mathematics and Optimization A Wiley-Interscience Publication John Wiley & Sons, Inc, New York, 1990 [7] Y Kohayakawa, V Rödl, J Skokan, Hypergraphs, quasi-randomness, and conditions for regularity J Combin Theory Ser A 97 00, no, [8] J Solymosi, A note on a question of Erdős and Graham Combin Probab Comput , no, [9] W T Trotter, Combinatorics and partially ordered sets Dimension theory Johns Hopkins Series in the Mathematical Sciences Johns Hopkins University Press, Baltimore, MD,