Counting patterns in colored orthogonal arrays
aa r X i v : . [ m a t h . C O ] A p r COUNTING PATTERNS IN COLORED ORTHOGONAL ARRAYS
A. MONTEJANO AND O. SERRA
Abstract.
Let S be an orthogonal array OA ( d, k ) and let c be an r –coloring ofits ground set X . We give a combinatorial identity which relates the numberof vectors in S with given color patterns under c with the cardinalities ofthe color classes. Several applications of the identity are considered. Amongthem, we show that every equitable r –coloring of the integer interval [1 , n ] hasat least ( nr ) + O ( n ) monochromatic Schur triples. We also show that in anorthogonal array OA ( d, d − Introduction
Arithmetic Ramsey Theory can be seen as the study of the existence of monochro-matic structures, like arithmetic progressions or solutions of linear systems, in everycoloring of sets of integers. The early results in the area are the theorem of Schuron monochromatic solutions of the equation x + y = z , the Van der Waerden theo-rem on monochromatic arithmetic progressions or the common generalization of thetwo, Rado’s theorem, on monochromatic solutions of linear systems. Anti–Ramseyresults refer to the study of combinatorial structures with elements of pairwise dis-tinct colors, or rainbow structures, a subject started by Erd˝os, Simonovits and S´os[6] which has received much attention since then. Canonical Ramsey theory collectsresults ensuring the existence of either a monochromatic or a rainbow structure.Jungi´c, Licht, Mahdian, Neˇsetˇril and Radoiˇci´c [9] started what they call RainbowRamsey Theory, which concerns the study of rainbow structures in colourings ofsets of integers.Counting versions of Arithmetic Ramsey results have also been obtained. Frankl,Graham and R¨odl [7] prove that, in a finite coloring of an integer interval, actuallya positive fraction of all solutions of a partition regular system are monochromatic.They also prove that, for every coloring of an integer interval, a positive fraction ofall solutions of such a system are either monochromatic or rainbow.Some of the above phenomena on the existence and number of color patterns incombinatorial structures behave in a particularly nice way when considered in finitegroups. A simple example is the fact that the total number of monochromatic Schurtriples in every two–coloring of the group of integers modulo n depends only on thecardinality of the color classes but not on the distribution of the colors, a fact firstnoticed, as far as we know, by Datskowsky [5]. The same is true for monochromaticthree–term arithmetic progressions when n is relatively prime with 6, as noted by Croot [4]. In [3] a combinatorial counting argument was given which explainsthe above two results and provides the ground for further generalizations in threedirections. First, results like the above mentioned ones can be extended to generalfinite groups. Actually the universe to be colored needs not to be even a group, butsimply the base set of an orthogonal array. Second, the monochromatic structuresinclude Schur triples, arithmetic progressions, or solutions of more general equationsin groups. Third, the counting argument can be applied to colorings with morethan two colors and can also be used to study rainbow structures or specific colorpatterns. Of course there are limitations in such general results, which become lessprecise with the increasing complexity of the structures we consider.We give in Section 2 a general formulation of the basic counting lemma (Lemma1) which is based in a counting argument used in [3]. Section 3 collects somespecific applications for orthogonal arrays OA (3 , ax + by + cz = d in an abelian group of order coprime with a , b and c or, more generally, equations of the form x α y β z γ = b in a group G , where α, β, γ are automorphisms of G . In this case Lemma 1 leads to a relationship betweenmonochromatic and rainbow triples which depends only on the cardinalities ofthe color classes (Theorem 2). This relationship provides results on the minimumnumber of monochromatic or of rainbow triples in orthogonal arrays (Corollary 3and Corollary 6).In the general context of orthogonal arrays OA (3 ,
2) the relationship betweenmonochromatic and rainbow triples can not be strengthened as illustrated in Ex-ample 4. Section 4 particularizes to linear equations of the form ax + by + cz = d in an abelian group. In particular Theorem is used to obtain a lower bound on thenumber of monochromatic Schur triples in an equitable r –coloring of the integerinterval [1 , n ] (Theorem 8). For 3–term arithmetic progressions, colorings whichare rainbow–free have been characterized in [11]. This characterization shows thatevery coloring with smaller color class sufficiently large has a rainbow triple. Herewe obtain a similar result for general groups non necessarily abelian (Corollary 9).It is also shown that every equitable coloring of an abelian group has at least alinear number of rainbow solutions of any given linear equation which is partitionregular (Corollary 10).Section 5 is devoted to orthogonal arrays of the form OA ( d, d − OA ( d, k )in Section 6 where Lemma 1 is used to show that, for every r –coloring of thebase set of an orthogonal array OA ( d, k ), a positive proportion of all vectors haspatterns in every ball of radius ( d − k ), where we use the ℓ distance in the set of r –vectors identifying color patterns (Theorem 13). This result provides a quantitativeestimation which is particularized in the case of almost monochromatic (all but at OUNTING PATTERNS IN COLORED ORTHOGONAL ARRAYS 3 most one entry of the same color) or almost rainbow (all but at most one of theentries pairwise distinct) patterns (Theorem ?? ).2. A counting argument
Let X be a finite set with cardinality n and let S be a set of vectors in X d . Let c : X → [1 , r ] be an r –coloring of X with color classes X , . . . , X r . A vector x = ( x , x , . . . , x d ) ∈ S is monochromatic under c if all its coordinates belong tothe same color class. When there are either no two coordinates of the same colorclass or all colors are present, we say that the vector is rainbow under c . We denoteby M = M ( S ) and R = R ( S ) the set of monochromatic and rainbow vectors in S respectively.A set S of d -vectors with entries in X is an orthogonal array of degree d andstrength k if, for any choice of k columns, each k -vector of X k appears in exactlyone vector of S . In other words, if we specify any set of k entries a , · · · , a k andany set of subscripts 1 ≤ i < i < · · · < i k ≤ d , we find exactly one vector y = ( y , y , . . . , y d ) in S with y i = a , y i = a , . . . , y i k = a k . We denote by OA ( d, k ) the family of orthogonal arrays of degree d and strength k on X .Lemma ?? below is the basic tool we shall use. It is based on the counting argumentsused in [3].In what follows we use the following notation. The color classes of an r –coloring of X will be denoted by X , X , . . . , X r , and we denote by c i = | X i | /n the density ofthe i -th color class. For a vector u = ( u , . . . , u r ) with nonnegative integer entries,we denote by | u | = P i u i . The multinomial coefficient (cid:0) du ,u ,...,u r ,d −| u | (cid:1) will bewritten as (cid:0) d u (cid:1) . For a vector v = ( v , . . . , v r ) we write (cid:0) vu (cid:1) = (cid:0) v u (cid:1)(cid:0) v u (cid:1) · · · (cid:0) v r u r (cid:1) . Weuse the convention (cid:0) vu (cid:1) = 0 if v < u and (cid:0) (cid:1) = 1. Lemma 1.
Let S be an orthogonal array OA ( d, k ) on X and let c be an r –coloringof X .For each vector u = ( u , u , . . . , u r ) with | u | ≤ k the following equality holds: (1) 1 n k X | v | = d (cid:18) vu (cid:19) s ( v ) = (cid:18) d u (cid:19) c u · · · c u r r , where the sum is extended to all vectors v = ( v , v , . . . , v r ) with nonnegative integerentries, | v | = d , and s ( v ) is the number of vectors in S with v i coordinates in X i for each i = 1 , . . . , r .Proof. Given an ordered partition V = ( V , V , . . . , V r ) of [1 , d ], possibly with someempty parts, let us denote by S ( V ) the set of vectors in S whose entries in V i belong to X i , 1 ≤ i ≤ r . An r -tuple of subsets ( U , U , . . . , U r ) of [1 , d ] is of type u = ( u , u , . . . , u r ) if | U i | = u i , 1 ≤ i ≤ r . Denote by P r ( u ) the set of all r –tuplesof pairwise disjoint subsets of [1 , d ] of type u . We say that V dominates U , andwrite V (cid:23) U , if V i ⊃ U i , 1 ≤ i ≤ r . A. MONTEJANO AND O. SERRA
Since S is an orthogonal array OA ( d, k ), there are k − | u | vectors in S whichmeet a prescribed assignment of | u | coordinates. Hence, for each r –tuple of subsets( U , U , . . . U r ) in P r ( u ) there are | X | u | X | u · · · | X r | u r | X | k −| u | vectors in S whoseentries in U i belong to X i , 1 ≤ i ≤ r . Among these vectors we find all vectors in S ( V ) for each partition V which dominates U , that is, X V (cid:23) U S ( V ) = | X | u | X | u · · · | X r | u r | X | k −| u | . Each partition V dominates (cid:0) | V | u (cid:1)(cid:0) | V | u (cid:1) · · · (cid:0) | V r | u r (cid:1) r –tuples in P r ( u ). Summing upthrough all r –tuples in P r ( u ) we get vectors counted by s ( v ) for each v whichdominates componentwise the vector u : (cid:18) d u (cid:19) | X | u | X | u · · · | X r | u r | X | k −| u | = X U ∈P r ( u ) X V (cid:23) U S ( V )= X V (cid:23) U X U ∈P r ( u ) S ( V )= X | v | = d (cid:18) v u (cid:19)(cid:18) v u (cid:19) · · · (cid:18) v r u r (cid:19) X V ∈P r ( v ) S ( V )= X | v | = d (cid:18) v u (cid:19)(cid:18) v u (cid:19) · · · (cid:18) v r u r (cid:19) s ( v ) . Dividing by n k we get equation (1). (cid:3) Lemma 1 gives a relationship between the number of vectors with some specificcolor patterns and the cardinalities of the color classes. This identity may providesome precise formulas for the number of vectors with a particular color pattern, orat least approximate counting results of a general nature. In the remaining of thepaper we give some applications of these identities.3.
Colour patterns in OA (3 , O (3 ,
2) we get a nice relationship between monochromaticand rainbow vectors.
Theorem 2.
Let S be an orthogonal array OA (3 , on X and n = | X | . For any r –coloring of X we have (2) 2 | M | − | R | = n (3 r X i =1 c i − , where M and R denote the set of monochromatic and rainbow vectors of S respec-tively.Proof. By taking u = (0 , ,
0) in Lemma 1 we get(3) | X | = X k v k =3 s ( v ) = | M | + | R | + | T (2 , | , OUNTING PATTERNS IN COLORED ORTHOGONAL ARRAYS 5 where T (2 ,
1) = S \ { M ∪ R } denotes the set of vectors in S with exactly two entriesof the same colour.On the other hand, the choice of u = (2 , ,
0) in Lemma 1 gives3 | X | = 3 s (3 , ,
0) + s (2 , ,
0) + s (2 , , . Adding up similar countings with (0 , ,
0) and (0 , , r X i =1 | X i | = 3 | M | + | T (2 , | . The result follows by substracting (3) from (4). (cid:3)
As an immediate consequence of Theorem 2 we get:
Corollary 3.
Let c be an r –coloring of the base set of an orthogonal array OA (3 , with α c = 3 P ri =1 c i − . If α c > then there are at least α c n monochromatic triplesand, if α c < , then there are at least | α c | n rainbow triples.In particular, every equitable coloring with r ≥ colors has at least (1 − /r ) n rainbow triples. In the context of orthogonal arrays there are examples which show that essentiallyall solutions for | M | and | R | in equation (2) are possible values for the numberof monochromatic and rainbow vectors in an orthogonal array whose points arecolored. We illustrate this fact with the following example. Example 4.
Let Y be a multiplicative quasigroup (we only require the cancellationlaw) and consider the quasigroup X = Y × Z with ( x, i ) ∗ ( y, j ) = ( xy, i + j )).The set of triples { ( x, i ) , ( y, j ) , ( x, i ) ∗ ( y, j ) : ( x, i ) , ( y, j ) ∈ X } is an orthogonalarray. The coloring χ ( x, i ) = i on X has the maximum possible number 3 | Y | ofmonochromatic triples and the maximum possible number 6 | Y | of rainbow triplesfor an equitable coloring of X .Let L be the latin square on X with entries L (( x, i ) , ( y, j )) = ( x, i ) ∗ ( y, j ) for each( x, i ) , ( y, j ) ∈ X . Let U, V ⊂ Y be subsets of Y . Exchange the entries in L of theform ( xy,
0) with ( xy,
1) for every x ∈ U and y ∈ V . The resulting orthogonal arrayhas 3 | Y | − | S | · | T | monochromatic triples for the same coloring of X .Let L ′ be the latin square obtained by the above procedure with U = V = Y .For each pair U ′ , V ′ ⊂ Y we can now exhange the entries ( xy,
0) with ( xy, x ∈ U ′ and y ∈ V ′ . The resulting orthogonal array has | Y | − | U ′ | · | V ′ | monochromatic triples for the same coloring. By choosing U ′ = V ′ = Y , thereare no monochromatic, and therefore no rainbow, triples. These are examples ofequitable colorings in orthogonal arrays for each value of | M | ∈ [0 , | Y | ] ∪ ( | Y | +2 · [0 , | Y | ]). (cid:3) For two–colorings there are no rainbow triples, so that Theorem 2 gives a formulafor the total number of monochromatic triples in terms of the cardinalities of the
A. MONTEJANO AND O. SERRA color classes. By minimizing that formula (with each color class of density 1 /
2) weget the minimum number of monochromatic triples in an orthogonal array OA (3 , X . More precisely, we have the next Corollarywhich is a natural generalization of Corollary 3.1 in [3], Corollary 5.
Let S be an orthogonal array OA (3 , on X . For any 2-coloring of X we have | M | = | X | − | X | · | X | + | X | . In particular, for any –coloring of X , there are at least n / monochromatic triplesin S . In the case of three–colorings Theorem 2 has a nice interpretation. Let us call σ c = r X i =1 c i /r − ( r X i =1 c i /r ) the variance of an r –coloring c . For r = 3 the expression on the right of equation(2) coincides, up to a constant, with the variance of the coloring (in particular isalways nonnegative.) Theorem 2 can be restated for three–colorings in the followingform. Corollary 6.
Let S be an orthogonal array OA (3 , on X . For any 3-coloring of X we have (5) 2 | M | − | R | = 9 σ c n . In particular, there are at least (9 σ c / n monochromatic triples. Linear equations
Natural extensions of results in Arithmetic Ramsey Theory concern the study ofcolor patterns of structures in groups. The results in Section 2 can be directlyapplied to this setting. The set of solutions of a linear equation of the form ax + by + cz = d in an abelian group of order coprime with a, b and c forms an orthogonal array OA (3 , x + y + z = − n ≡ t ) for some t ≥
1. Consider the partition A i = [0 , ( n/ t ) −
1] + i ( n/ t ), 0 ≤ i ≤ t −
1. We have A i + A i = [0 , n/ t ) −
2] +2 i ( n/ t ) and − A i − t − n/ (3 t ) , n − − i ( n/ t ), which are disjoint for each i , so that there are no monochromatic triples for that equation. Thus the lowerbound for rainbow triples given in Corollary 3 is also best possible. Moreover, thesame example for t = 1 shows that, for α c = 0, there are colorings of orthogonalarrays which have no monochromatic, and hence no rainbow, triples. OUNTING PATTERNS IN COLORED ORTHOGONAL ARRAYS 7
The lower bound on the number of monochromatic triples in Corollary 6 is also bestpossible. Consider for example Schur triples in a group G , triples of the form ( x, y, z )with xy = z . The set of Schur triples in a finite group forms an orthogonal array OA (3 , , n ] contains a rainbow Schur-triple. The result was later improvedby Sch¨onheim [15] who proved that any 3-coloring of the integers in [1 , N ] suchthat the smallest color class has more than N/ n/ Example 7.
Let
K < H < G be two subgroups of a finite group G such that K has index two in H and H has index two in G . Give color 1 to the elements in K , color 2 to the elements in H \ K and color by 3 the remaining elements of thegroup. In this example X X = X and X X = X X = X . Thus there are norainbow Schur triples under this coloring. (cid:3) It is not clear to us that the lower bound n/ , n ] is n /
11 + O ( n ). These authors exhibit a coloring with color classesof density 6 /
11 and 5 /
11 which attains the lower bound. The same result wasobtained by Schoen [13] and Datskovsky [5]. The later author used Corollary 5 asan intermediate step of his proof. By using Theorem 2 one can obtain a simpleproof of a lower bound on the number of Schur triples in an equitable coloring of[1 , n ] with an arbitrary number of colors.
Theorem 8.
Any equitable r –coloring of the integer interval [1 , n ] has at least | M | ≥ (1 / r ) n + O ( n ) monochromatic Schur triples.Proof. Let N = 2 n and consider the ( r + 1) coloring { X , X , . . . , X r , X r +1 } of thecyclic group Z /N Z where { X , X , . . . , X r } is the given three–coloring of [1 , n ] and X r +1 = [ n +1 , n ] (we identify the integers in [1 , n ] with its representatives modulo N ). We consider the Schur triples of Z /N Z ordered as ( x, y, z ) with x + y = z .By (2) we have2 | M ′ | − | R ′ | = (3 r +1 X i =1 c i − n ) = (3( r (1 / r ) + (1 / ) − n = − (1 − /r ) n , (6)where M ′ , R ′ are the sets of monochromatic and rainbow Schur triples respectively,and c i = | X i | / n are the densities of the color classes.There are | M X r +1 | = n / O ( n ) Schur triples of color X r +1 . The number P ri =1 | M X i | of monochromatic Schur triples of the other colors coincides, up to A. MONTEJANO AND O. SERRA O ( n ) terms, with the number | M | of monochromatic triples in the given coloringof the integer interval [1 , n ].Let us estimate the number | R ′ | of rainbow triples. For each u ∈ Y , Y ∈ { X , . . . , X r } ,we have the triples ( u, w − u, w ) , w ∈ [1 , u ] \ Y, and the triples ( u − w, w, u ) , w ∈ [ u, n ] \ Y. Therefore, for each u ∈ [1 , n ] there are (1 − /r ) n + O (1) such rainbow triples (eachcounted twice according to the permutation of the first two coordinates) giving riseto (1 − /r ) n + O ( n )rainbow Schur triples with the third coordinate in ∪ ri =1 X i . On the other hand, foreach u ∈ Y , Y ∈ { X , . . . , X r } , there also the rainbow Schur triples of the form( n − u, w, n + w − u ) and ( w, n − u, n + w − u ) , w ∈ [ u, n ] \ Y with the third coordinate in X r +1 . There are at least ( r − n/r − u + O (1) choicesfor such w , giving a total of at least2 ( r − n/r X u =1 (( r − n/r − u + O (1)) = 2(1 − /r ) n − (1 − /r ) n + O ( n )= (1 − /r ) n + O ( n )such rainbow triples. By plugging this estimation in (6) we get | M | ≥ (cid:0) ((1 − /r ) + (1 − /r ) − − (1 − /r )) n + O ( n ) (cid:1) = (1 / r ) n + O ( n ) . (cid:3) Let us consider next 3–term arithmetic progressions. Let G be a finite group anddenote by p ( G ) the smallest prime divisor of | G | . A d -term arithmetic progressionin a finite group G with p ( G ) ≥ d is a set of the form { a, ax, ax , . . . , ax k − } where a, x ∈ G . When G is abelian the set AP (3) of 3–term arithmetic progressionscorrespond to solutions of the equation x − y + z = 0.By proving a conjecture in [9] it was shown in [ ? ] that a 3–coloring of an abeliangroup G of order n such that the smaller color class has cardinality at least n/ p ( G )does have rainbow AP (3), and there are three–colorings of abelian groups in whichthe smallest color class has density 1 / AP (3). For a nonnecessarily abelian group G the following can be proved. Corollary 9. A –coloring of a group G with p ( G ) > with smaller color classof cardinality αn has at least (6 α (2 − α ) − / n rainbow AP (3) . In particular,if α > (0 . n then there is a rainbow AP (3) .Proof. By Corollary 6, the number | R | of rainbow AP (3) satisfies(7) | R | = 2 | M | − σ c n . OUNTING PATTERNS IN COLORED ORTHOGONAL ARRAYS 9
We have9 σ c = 3 X i =1 c i − ≤ α + (1 − α ) ) − α − α + 2 = 6 α (2 α −
3) + 2 . For a group G with p ( G ) >
53, it is shown in [3] that every 3–coloring of G has atleast n /
30 monochromatic AP (3). By substitution in (7) we get | R | ≥ (1 /
15 + 6 α (3 − α ) − n = (6 α (2 − α ) − / n . The last part of the statement follows since the coefficient of n in the above equa-tion is positive if α > (0 . n . (cid:3) The equations x + y − z = 0 and x − y + z = 0 for Schur triples and 3–term arithmeticprogressions in abelian groups are examples of regular equations, namely, equationsof the form ax + by + cz = 0 such that the sum of a nonempty subset of thecoefficients is zero. As another consequence of Corollary 6 we have the followingresult concerning rainbow solutions of such equations. Corollary 10.
Let ax + by + cz = 0 be a regular equation in an abelian group G oforder n . For every equitable –coloring of G there are at least n rainbow solutionsof the equation.Proof. If a + b + c = 0 then the system has the n solutions { ( x, x, x ) : x ∈ G } . If a + b = 0 then the system has the n solutions { ( x, − x,
0) : x ∈ G } . For an equitable3–coloring we have σ c = 0. Therefore Corollary 6 gives | R | ≥ | M | ≥ n . (cid:3) Color patterns in OA ( d, d − OA ( d, d −
1) with arbitrary d ≥
3, Lemma 1 gives the followingrelation.
Theorem 11.
Let S be an orthogonal array OA ( d, d − on a set X with cardinality n . For each r –coloring of X and each color class X i we have | S i | + ( − d − | M i | = ((1 − c i ) d − ( − d c di ) n d − , where S i denotes the set of vectors in S which miss color i and M i is the set ofmonochromatic vectors of color i . In particular, the total number | M | of monochro-matic vectors satisfies (8) r X i =1 | S i | + ( − d − | M | = r X i =1 (cid:0) (1 − c i ) d − ( − d c di (cid:1) n d − . Proof.
Without loss of generality we may assume i = 1. Consider the alternatingsum of the equations (1) for vectors of type u j = ( j, , . . . , , j = 0 , , . . . , d − We have d − X j =0 ( − j (cid:18) dj (cid:19) | X | j | X | d − − j = d − X j =0 ( − j X k v k = d (cid:18) v j (cid:19) s ( v )= X k v k = d d − X j =0 ( − j (cid:18) v j (cid:19) s ( v )= X k v k = d,v =0 s ( v ) + ( − d +1 s ( d, , . . . , | S | + ( − d +1 | M | , (9)where S denotes the set of vectors which miss color 1 and M denotes the set ofvectors with all entries of color 1. The first term of the above equalities can bewritten as 1 n (cid:0) ( n − | X | ) d − ( − d | X | d (cid:1) = (cid:0) (1 − c ) d − ( − d c d (cid:1) n d − , which gives the first part of the statement. Equality (8) is simply obtained byadding up the equations (9) for i = 1 , . . . , d . (cid:3) For 2–colorings, S and S are just the set of monochromatic vectors of color 2 and 1respectively. Thus equation (8) shows that, for d odd, the number of monochromaticvectors depends only on the cardinalities of the color classes independently of theirdistribution, which is the main result in [3].In particular we get the following Corollary for 3–colorings, which is a slight gen-eralization of a result by Balandraud [2, Corollary 2]. Here a vector is said to be rainbow if all colors are present. Corollary 12.
Let S be an orthogonal array OA ( d, d − on X . For each –coloringof X we have (1 + ( − d − ) | M | − | R | = X i =1 ((1 − c i ) d − ( − d c di ) − ! n d − . Proof.
With our current notion of rainbow vectors we have | R | = | ∩ j =1 ¯ S j | = | S | − X i =1 | S i | + | M | . By substitution in the last equation of Theorem 11 we have X i =1 | S i | + ( − d +1 | M | = | X | d − − | R | + (1 + ( − d +1 ) | M | = r X i =1 (cid:0) (1 − c i ) d − ( − d c di (cid:1) n d − , as claimed. (cid:3) OUNTING PATTERNS IN COLORED ORTHOGONAL ARRAYS 11
It follows from Corollary 12 that, for d even, the number of rainbow vectors in a3–coloring of the base set of an orthogonal array OA ( d, d −
1) depends only on thecardinality of the color classes but not on the distribution of the colors.6.
Color patterns in OA ( d, k )Orthogonal arrays OA ( d, k ) include sets of solutions of linear systems: for a ( d × m )integer matrix A such that every ( m × m ) submatrix is nonsingular, the set ofsolutions of the linear system Ax = b in an abelian group forms an orthogonal array OA ( d, d − m ). In this general context Lemma 1 still provides some information onthe distribution of color patterns. Recall that a color pattern in an r –coloring of thebase set of an orthogonal array OA ( d, k ) is identified by a vector v = ( v , . . . , v r )with P i v i = d where entry v i denotes the number of appearances of color i .We consider the distance between two color r –vectors u , w given by d ( u , w ) = P i | u i − w i | . Theorem 13.
Let S be an orthogonal array OA ( d, k ) on a set X and let c be an r –coloring of X with α = min i c i . For each color pattern v = ( v , . . . , v r ) there isa color pattern v ′ = ( v ′ , . . . , v ′ r ) at distance d ( v , v ′ ) ≤ d − k ) such that there areat least s ( v ′ ) ≥ (cid:0) d − k + r − r − (cid:1) ( αn ) k vectors of S colored with v ′ .Proof. We say that an r –vector w dominates the r –vector u , written w (cid:23) u , if w ≥ u , . . . , w r ≥ u r .Let v be a given color pattern and choose u = ( u , . . . , u r ) with | u | = k such that v (cid:23) u . For every vector v ′ with | v ′ | = d we have (cid:0) v ′ u (cid:1) ≤ (cid:0) d u (cid:1) . By equation (1), (cid:18)(cid:18) d u (cid:19) c u · · · c u r r (cid:19) n k = X v ′ (cid:23) u (cid:18) v ′ u (cid:19) s ( v ′ ) ≤ (cid:18) d u (cid:19) X v ′ (cid:23) u s ( v ′ ) . There are at most (cid:0) d − k + r − r − (cid:1) vectors v ′ with | v ′ | = d which dominate u , and eachsuch v ′ is at distance d ( v ′ , v ) ≤ d ( v ′ , u ) + d ( v , u ) ≤ d − k ) from v . Hence, if α = min { c , . . . , c r } ,max v ′ (cid:23) u s ( v ′ ) ≥ (cid:0) d − k + r − r − (cid:1) ( c u · · · c u r r ) n k ≥ (cid:0) d − k + r − r − (cid:1) ( αn ) k . (cid:3) References [1] V. E. Alekseev and S. Savchev. Problem M. 1040. Kvant, (1987) 4:23.[2] E. Balandraud, Coloured Solutions of Equations in Finite Groups,
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