An EGZ generalization for 5 colors

Transcription

1 An EGZ generalization for 5 colors David Grynkiewicz and Andrew Schultz July 6, 00 Abstract Let g zs(, k) (g zs(, k + 1)) be the inial integer such that any coloring of the integers fro U U k 1,..., g zs (, k) by i=1z i (the integers k fro 1,..., g zs (, k + 1) by i=1z i { }) there exist integers such that x 1 < < x < y 1 < < y 1. there exists j x such that (x i ) Z P j x for each i and i=1 x i = 0 od (or (x i) = for each i);. there exists j y such that (y i) Z P jy for each i and i=1 yi = 0 od (or (y i ) = for each i); and 3. (x x 1 ) y x 1. In this note we show g zs (, ) = 5 4 for, g zs (, 3) = for 4, gzs (, 4) = 10 9 for 3, and g zs (, 5) = 13 for. 1 Introduction Denote by [a, b] the set of integers x such that a x b. For a set S, an S-coloring of [a, b] is a apping : [a, b] S. If S = {1,..., r}, we say is an r-coloring. The following is the Erdős-Ginzburg-Ziv Theore, [9] [8]. Proposition 1.1. Any sequence of at least 1 eleents of Z contains a subsequence of eleents whose su is zero odulo. Several theores of Rasey-type have been generalized by considering Z - colorings and zero-su configurations rather than -colorings and onochroatic configurations. Such theores are called generalizations in the sense of Departent of Matheatics, Caltech, Pasadena, CA, 9115 Supported by NSF grant DMS

2 EGZ. Best known of these results is the zero-su-tree theore [4] [19], and other results concerning graphs and hypergraphs can be found in [10] and [1]. Rasey-type probles dealing with colorings of the natural nubers can be classified as one-set probles initiated in [7] and further explored in [3] [6] [15] [16] [17] [18] and two-set probles initiated in [5] and further investigated in [13] [0] [1]. We introduce soe definitions towards generalizing two-set probles in the sense of EGZ. Let the set k i=1 Zi denote the pairwise disjoint union of k copies of the set of eleents Z and let denote a sybol such that / k i=1 Zi. Just as the EGZ Theore generalizes the pigeonhole principle for boxes and pigeons, the following observation generalizes to an arbitrary nuber of boxes. Observation 1.. Let and r = k (r = k + 1) be positive integers. Any sequence of at least r( 1) + 1 eleents fro k i=1 Zi (fro k i=1 Zi { }) contains a subsequence of eleents fro soe Z i whose su is zero odulo (or a subsequence of eleents). For a positive integer r and a syste of inequalities L in variables, let R(L; r) denote the inial integer N such that every r-coloring of [1, N] contains two sets S 1 and S, each being onochroatic and of cardinality, such that S 1 S fors a solution to L. In a siilar way, if r = k (r = k + 1) let R zs (L; r) denote the iniu integer N such that every k i=1 Zi -coloring ( k i=1 Zi { }-coloring) of [1, N] contains two sets S 1 and S, each being zero-su in Z i (or -onochroatic) and of cardinality, such that S 1 S fors a solution to L. It is easy to see that R(L; r) R zs (L; r). If equality holds for a given r, we say the syste L adits an EGZ generalization for r colors. Using the definitions above, it was proved in [5] that R( L; ) = 5 3 and R( L; 3) = 9 7 where L := x 1 < x < < x < y 1 < y < < y ; x x 1 y y 1. Furtherore, they proved L adits an EGZ generalization for and 3 colors. Recently in a sequence of three papers [13], [1], [11] the first author showed R( L; 4) = 1 9 and that L adits an EGZ generalization for 4 colors. In achieving this result, a new tool was developed in [11]. We state in Proposition.1 a particular case equivalent to a result fro [14]. It sees that the deterination of R( L; 5) would not be easy or short. At present the authors are not aware of any nontrivial two set EGZ generalizations for 5 colors.

3 The otivation for this paper is twofold. First, we wished to find a syste L that adits an EGZ generalization for 5 colors. Second, we wanted to test the conjecture below fro []. Conjecture 1.3 Let k and be positive integers. If L 1 and L are two systes of inequalities in variables such that every positive integer solution of L 1 is a solution of L, and if L adits an EGZ generalization in r colors, then L 1 adits an EGZ generalization in r colors. Toward these ends we have chosen to look at the syste L first investigated by the second author in [0] defined by L := x 1 < x < < x < y 1 < y < < y ; y x 1 (x x 1 ). In Section we state soe preliinary definitions and tools, and in Section 3 we deterine R zs (L; r) for r [, 5]. In conjunction with the results fro [0], these results show L adits EGZ generalizations for these values of r. Preliinaries Along with the EGZ theore we shall need the following result, an easy consequence of the EGZ Theore and [11] or [14]. Proposition.1. If H = (h 1,..., h k ) is a sequence of at least 1 eleents fro Z, then one of the following holds: 1. any sequence of 3 eleents fro H contains a subsequence of eleents whose su is zero odulo. there exists a partition {A i } k i=1 of H \ {h k } with k i=1 A i = ; or 3. there exists j Z such that h i = j for all but eleents h i H. An -set, denoted Z = (z 1,..., z ), is a sequence of distinct positive integers such that z 1 < < z. For a pair of -sets X and Y, we write X Y if x < y 1. For Z = (z 1,..., z ) we also adopt the following notation: (i) int i (Z) = z i for i ; (ii) first k (Z) = {z 1,..., z in{k,} }; (iii) first(z) = z 1 ; (iv) last k (Z) = {z ax{1,n k},..., x }; and (v) last(z) = z. 3

4 For atters of notation and consistency with [0], we shall denote R zs (L, r) by g zs (, r). To facilitate our evaluation of g zs (, r), we ake the following observation. Observation.. Let positive integer r = k (integer r = k +1) be given, and let : [1, n] k i=1 Zi ( : [1, n] k i=1 Zi { }) be given. If there exists a zero-su (zero-su or onochroatic) -set Y [r( 1) + 1, n] such that y r( 1) + 1, then the syste L is satisfied. Proof. By Observation 1. there is soe zero-su or onochroatic -set X [1, r( 1) + 1]. If a zero-su or onochroatic -set Y [r( 1) + 1, n] exists, then X Y. If y r( 1)+1 we have y x 1 r( 1)+1 x 1 (r( 1) + 1 x 1 ) (x x 1 ). 3 Evaluation of g zs (, r) for r [, 5] The deterination of g zs (, ) is a siple application of the EGZ Theore. Theore 3.1. If is an integer, then g zs (, ) = 5 4. Proof. That g zs (, ) 5 4 follows fro g(, ) = 5 4 as found in [0] and the trivial fact that g zs (, ) g(, ). By Observation. it is sufficient to find a zero-su -set Y [, 5 4] with y 4 3. Let P = [3, 5 4]. Since P = 1 there exists soe zero-su -set Y P. Since P [3, 4 4] = 1 it follows that y 4 3. The deterination of g zs (, 3) will require the use of Proposition.1 and the following two leas, proofs for which can be found in [0]. Lea 3.. Let 4 be an integer, and let : [1, 3 4] Z { } be a coloring. If 1 ( ) 3, then there exist onochroatic -sets X Y such that y x 1 (x x 1 ). Lea 3.3. Let 4 be an integer. If : [3 1, 7 + 6] [1, 3] is a given coloring, then either 1. there exists a onochroatic -set Y such that y 6 5 or. there exist onochroatic -sets W Y such that y w 1 (w w 1 ). Theore 3.4. If 4 is an integer, then g zs (, 3) =

5 Proof. That g zs (, 3) follows fro g(, 3) = as found in [0] and the trivial fact that g zs (, 3) g(, 3). 6. Let : [1, 7 + 6] Next we show that g zs (, 3) 7 + Z { } be an arbitrary coloring. By Observation. it is sufficient to find a zero-su or onochroatic -set Y [3 1, 7+ For convenience we let P = 1 (Z ) [3 1, 7 + 6]. 6] with y 6 5. To coplete the proof we consider three cases based on k = 1 ( ) [6 5, 7 + 6]. Case 1. Suppose k = 0. If P 1, one ay find a zero-su set Y [3 1, 7 + 6] with y 6 5. Otherwise P < 1, so that 1 ( ) [3 1, 6 6] 3. Shifting [3 1, 6 6] to the interval [1, 3 4] and applying Lea 3. copletes the proof. Case. Suppose 0 < k, so that P [6 5, 7 + 6] = + k. If 1 ( ) [3 1, 6 6] 1, then we are done. Hence we ay assue otherwise, so that 1 (Z ) [3 1, 6 6]. Selecting P P such that P = 1 and P [6 7, 7+ 6] = copletes the case. Case 3. Suppose < k <. We ay assue that 1 ( ) [3 1, 6 6] < k, since otherwise the proof is coplete by taking Y = last ( 1 ( ) [3 1, 7 + 6]. Hence, P > 1. We coplete this case by considering three subcases. Subcase 3a. Suppose any 3 eleents of P contain a zero-su -set. Since P [6 5, 7 + 6] + 1 we ay select P P with 3 eleents such that P [6 5, 7 + 6] + 1. By assuption P contains a zero-su -set Y. Since P [3 1, 6 6] < it follows that y 6 5. P +1 Subcase 3b. Suppose there exists a partition {A i } i=1 of P where A P +1 = {last(p )} with P A i =. i=1 Hence there exists a zero-su -set Y with y = last(p ) 6 5. Subcase 3c. If neither Subcase 3a nor 3b apply, then by Proposition.1 it 5

6 follows that (p) = j Z for all but at ost eleents p P ; for convenience, let H = {p P (p) j}. Induce a coloring e : [3 1, 7 + 6] [1, 3] defined by 1, for x P \ H e (x) =, for x H 3, for (x) = Note that any onochroatic -set W in e is either a zero-su or onochroatic -set in since 1 (). Hence, the result follows by Lea 3.3. Case 4. If k the result follows trivially. We consider the evaluation of g zs (, 4). Towards that end we use the following lea, the proof for which ay be found in [0]. Lea 3.5. Let 3 be an integer. If : [4, 10 9] [1, 4] is a given coloring, then either 1. there exists a onochroatic -set Y such that y 8 7 or. there exist onochroatic -sets W Y such that y w 1 (w w 1 ). Theore 3.6. If 3 is an integer, then g zs (, 4) = Proof. That g zs (, 4) 10 9 follows fro g(, 4) = 10 9 as found in [] and the trivial fact that g zs (, 4) g(, 4). Next we show that g zs (, 4) Let : [1, 10 9] Z 1 Z be an arbitrary coloring. By Observation. it is sufficient to find a zero-su -set Y [4, 10 9] with y 8 7. Since [8 7, 10 9] = 1, without loss of generality we ay assue 1 (Z ) [8 7, 10 9] = + k where k 0. If 1 (Z ) [4, 8 8] 1 k, then by the EGZ theore there exists a zero-su -set Y [4, 10 9] with y 8 7. Hence we ay assue 1 (Z ) [4, 8 8] k. (1) Letting P = 1 (Z 1 ) [4, 10 9], we thus have P [4, 8 8] k. We finish the proof by considering three cases. Case 1. Suppose any 3 eleents of P contain a zero-su -set. If such an -set Y satisfies y 8 7 we are done; hence we assue that any 3 ( 1 k) = +k+1 eleents fro 1 (Z 1 ) [4, 8 8] contain 6

7 a zero-su -set. Hence, for W = first +k+1 ( 1 (Z 1 ) [4, 8 8]) there exists soe -set W W that is zero-su. By Equation 1 we have t = 1 (Z ) [4, w ] k, () so that w w 1 + k t 1. Hence, if there exists a zero-su -set Y such that W Y and y (w w 1 ) + w 1 ( + k + t) + 4 (3) we shall be done. Taking Y = last +k+1 ( 1 (Z 1 ) [4, 8 8]), we see that there exists an -set Y Y which satisfies these requireents as follows. + k + 1) any eleents First, it is quickly verified that there are at least ( fro 1 (Z 1 ) in [4, 8 8] for every 3. Hence, we have W Y, fro which it follows that W Y. Second, we note that y 8 8 Y \ Y ( 1 (Z ) [4, 8 8] t). By using Equations 1 and, one ay easily verify that this iplies y ( + k + t) + 4, so that Equation 3 is satified. P +1 Case. Suppose there exists a partition {A i } i=1 of P where where A P +1 = {last(p )} such that P +1 i=1 A i =. Hence there exists a zero-su -set Y [4, 10 9] with y = last(p ) 8 7. Case 3.If neither Case 1 nor Case apply, then by Proposition.1 it follows that (p) = j Z 1 for all but at ost 1 eleents p P ; for convenience, define H = {p P (p) j}. Induce a coloring e : [4, 10 9] [1, 4] defined by 1, for x P \ H, for x H e (x) = 3, for x first 1 ( 1 (Z 1 ) [4, 10 9]) 4, for x = int i ( 1 (Z 1 ) [4, 10 9]), i Note that any onochroatic -set X in e is either a zero-su -set in since 1 (j) for each j 1. Hence, by Lea 3.5 the proof is 7

8 coplete. The evaluation of g zs (, 5) requires two results fro the evaluation of g(, ) and g(, 5) found in [0]. Lea 3.7. Let be an integer. If : [5 3, 13 1] [1, 5] is a given coloring, then either 1. there exists a onochroatic -set Y such that y 10 9 or. there exist onochroatic -sets W Y such that y w 1 (w w 1 ). Lea 3.8. Let be an integer. If : [5 3, 10 10] [1, ] is a coloring with 1 (c) for soe c [1, ], then there exist onochroatic -sets X Y such that y x 1 (x x 1). Theore 3.9. If is an integer, then g zs (, 5) = Proof. That g zs (, 5) 13 1 follows fro g(, 5) = 13 1 as found in [0] and the trivial fact that g zs (, 5) g(, 5). We now show g zs (, 5) Let : [1, 13 1] Z 1 Z { } be an arbitrary coloring. By Observation. it is sufficient to find a zero-su -set Y [5 3, 13 1] with y The case of 1 ( ) [10 9, 13 1] is trivial, so we ay assue otherwise. Hence, it follows that 1 (Z i ) [10 9, 13 ] for soe i [1, ], say i =. Furtherore, if 1 ( ) [10 9, 13 1] = 0, then either 1 (Z ) [10 9, 13 1] 1 or 1 (Z 1 ) [10 9, 13 1]. In the forer case the proof is coplete by the EGZ Theore; in the latter case, we ust have ( 1 (Z 1 ) 1 (Z )) [5 3, 10 10]. Hence we ay induce a coloring e : [5 3, 10 10] [1, ] defined by { 1, for (x) = e (x) =, for x ( 1 (Z 1 ) 1 (Z )) [5 3, 10 10]. Since any onochroatic -set in e is also onochroatic in, the result follows fro by Lea 3.8. Hence 1 ( ) [10 9, 13 1] > 0 Let k = ( 1 ( ) 1 (Z )) [10 9, 13 1]. Clearly k < 3. We also have ( 1 ( ) 1 (Z )) [5 3, 10 10] < 3 k, (4) 8

9 since otherwise there exists either a zero-su or onochroatic -set Y with y [10 9, 13 1]. Likewise we ay assue that any 1 1 (Z 1 ) [10 9, 13 1] = 1 (3 k) = k + 1 eleents fro 1 (Z 1 ) [5 3, 10 10] contain a zero-su -set. Let W = first k+1 ( 1 (Z 1 ) [5 3, 10 10]), so that by assuption there exists a zero-su -set W W. Fro Equation 4 we see that t = ( 1 ( ) 1 (Z )) [5 3, w ] 3 3 k (5) so that w w 1 k t 1. Hence, if there exists a zero-su -set Y such that W Y and y (k + t) + x k + t 3 (6) we shall be done. Taking Y = last k+1 ( 1 (Z 1 ) [5 3, 10 10]), we see that there exists an -set Y Y which satisfies these requireents as follows. First, using Equation 4 one ay verify that there are at least (k +1 ) any eleents fro 1 (Z 1 ) in [5 3, 10 10] for every. Hence, we have W Y, fro which it follows that W Y. Second, we note that y Y \ Y ( ( 1 ( ) 1 (Z )) [5 3, 10 10] t). By using Equations 4 and 5, one ay verify that this iplies y 3 + k + t 3, so that Equation 6 is satified. Acknowledgent: The authors would like to thank Professor A. Bialostocki for suggesting that we investigate Conjectures 1.3 and for any fruitful discussions. References [1] P. Baginsky, A generalization of a Rasey-type theore on hyperatchings. Manuscript, 00. [] A. Bialostocki, Private Counication, June 00. [3] A. Bialostocki, G. Bialostocki, and D. Schaal, A zero-su theore. To appear in J. Cobin. Theory, Ser A. 9

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