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k-BARYCENTRIC OLSON CONSTANT

OSCAR ORDAZ, MAR´

IA TERESA VARELA and FELICIA VILLARROEL

Let Gbe a ﬁnite Abelian group of order n. A k-sequence in Gis said to be

barycentric if it contains an element which is the “average” of its terms. The k-

barycentric Olson constant BO(k, G) is introduced as the minimal positive integer

tsuch that any t-set in Gcontains a k-barycentric set. Conditions for the existence

of BO(k , G) are established and some values or bounds are given. Moreover,

new values for the k-barycentric Davenport constant BD(k , G) and barycentric

Ramsey numbers for stars BR(K1,k , G) are given. These constants, deﬁned in

some of our previous works, use sequences instead of sets.

AMS 2000 Subject Classiﬁcation: 05C55, 05C65, 05D10.

Key words: k-barycentric sequence, k-barycentric, k-barycentric Olson constant,

barycentric Ramsey number, zero-sum.

1. INTRODUCTION

Let Gbe an Abelian group of order n. Weighted sequences, i.e., sequences

constituted by terms of the form wiai, where aiare elements of Gand the coef-

ﬁcients or weights are positive integers, appear initially in the Caro conjecture

[2] formulated in 1996.

Conjecture 1.1.Let w1, w2, . . . , wkbe positive integers such that w1+

w2+· · ·+wk= 0 (mod n). Let a1, a2, . . . , an+k−1in Gnot necessarily distinct.

Then there exist kdistinct indices i1, . . . , iksuch that w1ai1+w2ai2+· · · +

wkaik= 0.

Hamidoune [11] gave a solution to Caro conjecture under the additional

assumption (wi, n) = 1 ∀i. Recently, Grynkiewicz [10] gave a complete solution

to this conjecture in

Theorem 1.2 ([10]).Let m, n and k≥2be positive integers. If fis a

sequence of n+k−1elements from a nontrivial Abelian group Gof order n

and exponent m, and if W={wi}k

i=1 is a sequence of integers whose sum is

zero modulo m, then there exists a rearranged subsequence {bi}k

i=1 of fsuch

that

k

P

i=1

wibi= 0.Furthermore, if fhas a k-set partition A=A1, . . . , Aksuch

MATH. REPORTS 11(61),1(2009), 33–45

34 Oscar Ordaz, Mar´ıa Teresa Varela and Felicia Villarroel 2

that |wiAi|=|Ai|for all i, then there exists a nontrivial subgroup Hof Gand

ak-set partition A1=A1

1, . . . , A1

kof fwith H⊆

k

P

i=1

wiA1

iand |wiA1

i|=|A1

i|

for all i.

A weighted k-sequence, k≥2,with terms wiai,wi= 1,1≤i≤k,

excepting some jfor which wj= 1 −k,ai∈Gand zero-sum, is called a

k-barycentric sequence. That is to say, a k-sequence ai, 1 ≤i≤k, with k≥2,

where there exists an ajsuch that a1+a2+· · · +aj+· · · +ak=kaj.

The study of barycentric sequences started in [6] and [7]. In [6] the

Davenport barycentric constant BD(G) and in [7] the Davenport k-barycentric

constant BD(k, G) are introduced. They are deﬁned as the smallest positive

integer tsuch that every sequence of length tcontains a barycentric sequence

or a k-barycentric sequence, respectively.

Let H= (V(H), E(H)) be a graph with e(H) edges. In [7] the barycen-

tric Ramsey number BR(H, G) is deﬁned as the minimum positive integer t

such that any coloring c:E(Kt)→Gof the edges of Ktby elements of G

yields a copy of H, say H0, with an edge e0such that

(1) X

e∈E(H0)

c(e) = kc(e0)

In this case His called a barycentric graph. This constant is a generalization of

the Ramsey zero-sum number R(H, G) deﬁned when e(H) = 0 (mod n), as the

minimal positive integer ssuch that any coloring c:E(Ks)→Gof the edges of

Ksby elements of Gcontains a copy of H, say H, with P

e∈E(H)

c(e) = 0, where 0

is the zero element of G. The necessity of the condition e(H) = 0 (mod n) for

the existence of R(H, G) is clear. It comes from the monochromatic coloration

of the edges of H.

The main goals of this paper are:

– To deﬁne of the k-barycentric Olson constant BO(k, G) and study its

existence. The Abelian groups that will be studied are G=Znfor some prime

or composite n, in particular for 3 ≤n≤12 and 3 ≤k≤n. For those kand

nfor which BO(k, Zn) exists, this value or bounds of it are given.

– To establish new values for BD(k, G) from BO(k, G).

– To give new values for BR(K1,k, G) from BD(k, G).

Besides this introduction and the conclusion, the paper contains three

main sections. Section 2 presents the tools that are used in Sections 3 and 4.

Section 3 is devoted to a study of the k-barycentric Olson constant BO(k, G)

for some kand G. In Section 4, new values of BD(k, G) using the values of

3k-barycentric Olson constant 35

BO(k, G) are established. Moreover, new values of BR(K1,k, G) from BD(k, G)

are given.

2. TOOLS

The following deﬁnition is introduced in [6] and [7] and constitutes a

natural extension of zero-sum sequences.

Deﬁnition 2.1.Let Abe a ﬁnite set with |A| ≥ 2 and Ga ﬁnite Abelian

group. A sequence f:A→Gis said to be barycentric if there exists a∈Asuch

that P

x∈A

f(x) = |A|f(a). The element f(a) is called barycenter. The sequence

is said to be k-barycentric when |A|=k. Moreover, when fis injective we use

barycentric set instead of barycentric sequence.

Hamidoune [11] gave the following condition.

Hamidoune-condition. Let Gbe an Abelian group of order n≥2 and

f:A→Ga sequence with |A| ≥ n+k−1. Then there exists a k-barycentric

subsequence of f. Moreover, in the case where k≥ |G|, the condition |A| ≥

k+D(G)−1, where D(G) is the Davenport constant, is suﬃcient for the

existence of a k-barycentric subsequence of f.

Deﬁnition 2.2.Let Gbe an Abelian group of order n≥2. The k-

barycentric Davenport constant BD(k, G) is the minimal positive integer t

such that every t-sequence in Gcontains a k-barycentric subsequence.

By Hamidoune-condition, we have BD(k, G)≤n+k−1.

In what follows, we present some results about orbits. More information

is available in [14].

The set Gn={fa,b :Zn→Zn, fa,b(x) = ax +b, a, b ∈Zn,(a, n) = 1}

is a group of order nφ(n) where φ(n) = |{0< q < n : (q, n) = 1}| is the

Euler phi-function. Let Xk

n={{x1, x2, . . . , xk}:xi∈Zn}.An action of Gn

on Xk

nis deﬁned as fa,b({x1, . . . , xk}) = {fa,b (x1), . . . , fa,b(xk)}.It is easy to

see that θ({0}) is the only orbit of X1

nand that the orbits of X2

nare of the

form θ({0, z}).

The Bezout theorem [5] and the Chinese remainder theorem allow to

formulate

Lemma 2.3.θ({0, z}) = θ({0, t}), with t= (z, n). Moreover, each orbit

in X2

ncontains one and only one {0, t}with t|nand t < n. There are as many

orbits in X2

nas there are divisors of nin {1,2, . . . , n −1}.

Remark 2.4.The orbits of Xk

ncan be obtained considering for each

orbit θ({x1, . . . , xk−1}) of Xk−1

nthe sets {x1, . . . , xk−1, xk}with xk∈Zn\

{x1, . . . , xk−1}. The action of Gnon these sets deﬁnes its orbits.

36 Oscar Ordaz, Mar´ıa Teresa Varela and Felicia Villarroel 4

We have the easy lemmas below.

Lemma 2.5.If {x1, . . . , xk}is a k-barycentric set, then all the elements

of θ({x1, . . . , xk})are k-barycentric. Such an orbit is called barycentric. More-

over, if {x1, . . . , xk}contains a t-barycentric set, then every element of θ({x1,

. . . , xk})also contains a t-barycentric set.

Lemma 2.6.The number of orbits of Xk

nis equal to the number of orbits

of Xn−k

n.

Proof. If x, y are in the same orbit of Xk

n, then Zn\xand Zn\yalso

are in the same orbit of Xn−k

n.Consequently, the bijection between the sets

Xk

nand Xn−k

noriginates a bijection between the orbits of Xk

nand Xn−k

n.

We also need the Dias da Silva-Hamidoune theorem below.

Theorem 2.7 ([8]).Let Hbe a subset of Zp. Let dbe an integer such

that 2≤d≤ |H|. Set ∧dH=nP

x∈S

x:S⊂H, |S|=do. Then | ∧dH| ≥

min{p, d(|H| − d)+1}.

We shall also use

Lemma 2.8 (Hamidoune [12]).Let Abe a subset of Znsuch that |A| ≥

n+3

2. Then ∧2A=Zn.

In what follows we present some results about decomposing a complete

graph into edge-disjoint subgraphs.

Theorem 2.9 (Harary [15]).Let Knbe a complete graph with nvertices.

Then Kn, with nodd, is the edge-disjoint union of n−1

2Hamiltonian cycles

while Kn, with neven, is the edge-disjoint union of n−2

2Hamiltonian cycles

and one perfect matching. Hence Kncan be decomposed into n−1perfect

matchings.

Corollary 2.10.Let Knbe a complete graph with nvertices, with n

odd. Then Kncan be decomposed into two complete graphs Kn+1

2

sharing a

vertex and a bipartite complete graph Kn−1

2,n−1

2

.

Corollary 2.11.Let Knbe a complete graph with nvertices, with n

even. Then Kncan be decomposed into two vertex-disjoint complete graphs

Kn

2and the remaining Kn

2,n

2into one perfect matching and one (n

2−1)-

regular graph.

Moreover, the following result is used to give the value of BR(K1,m, Zn)

with m= 0 (mod n).

5k-barycentric Olson constant 37

Theorem 2.12 (Bialostocki [1], Caro [4]).Let K1,m be the stars on m

edges with m= 0 (mod n).Then

BR(K1,m , Zn) = R(K1,m, Zn) = m+n−1if m=n= 0 (mod 2)

m+notherwise.

From now on, pwill be a prime number.

3. k-BARYCENTRIC OLSON CONSTANT

Let Gbe an Abelian group of order n≥3. We establish the existence

of BO(k, G) any derive some values or bounds. The existence of BO(k, Zp) is

established in Corollary 3.4 and Remark 3.2.

Deﬁnition 3.1.Let Gbe an Abelian group of order n≥3. The k-bary-

centric Olson constant BO(k , G) is the minimal positive integer tsuch that

every t-set in Gcontains a k-barycentric set, provided such an integer exists.

Remark 3.2.It is clear that when nis odd, BO(n, Zn) = nwhile if n

is even, BO(n, Zn) does not exist. Moreover, for nodd, the barycenter of

every Q⊆Znwith |Q|=n−1 is Zn\Q. Therefore, BO(n−1, Zn) does

not exist. For neven, BO(n−1, Zn) = n−1. Let S⊆Znwith |S|=n−1

and {b}=Zn\S. It is easy to see that Sis an (n−1)-barycentric set with

n

2−1

n−1bas barycenter.

Theorem 3.3 ([6]).Let s≥2,d≥2,p≥d+2+ 1

d−1. Let Abe a set

with s+delements, and f:A→Zpa sequence with |f(A)| ≥ p−1

d+d+ 1.

Then fcontains an s-barycentric subsequence.

Corollary 3.4.BO(k, Zp)≤pfor 3≤k≤p−2with p≥5.

Theorem 3.5.BO(k, Zn)≤nfor n≥6and n+1

2≤k≤n−2.

Proof. Let A⊆Znsuch that |A|=k+1. Then |A| ≥ n+3

2. By Lemma 2.8

there exists {u, v} ⊆ Asuch that u+v=P

A

x−ka +a, for some a∈Zn\A.

Then (A\ {u, v})∪ {a}is a k-barycentric set.

Lemma 3.6 ([7]).If a 3-sequence in Zn,nodd, is barycentric, then its

elements are equal or pairwise diﬀerent. Moreover, a 3-set in Znis barycentric

if and only if its elements are in arithmetic progression.

From this lemma we derive

Remark 3.7.BO(3, Zn)≤nfor n≥3.

The following result follows from the Dias da Silva-Hamidoune theorem.

38 Oscar Ordaz, Mar´ıa Teresa Varela and Felicia Villarroel 6

Theorem 3.8.BO(3, Zp)≤ dp

3e+ 1 for p≥5.

Proof. Let Abe a set in Zpwith |A|=dp

3e+1. By the theorem of Dias da

Silva-Hamidoune we have |∧2A| ≥ min{p, 2(|A| − 2)+1}= min{p, 2dp

3e−1}=

2dp

3e − 1. Set D={2x:x∈A}. Then | ∧2A|+|D| ≥ 3dp

3e> p. Therefore,

there exist 2x∈Dand y+z∈ ∧2Asuch that 2x=y+z, hence x+y+z= 3x,

i.e., {x, y, z}is a 3-barycentric set of A.

In particular, we deduce that

–BO(3, Z5) = 3;

– the set {0,4,6}shows that BO(3, Z7)≥4 so, by Theorem 3.8,

BO(3, Z7) = 4;

– the set {1,2,4,5}shows that BO(3, Z11 )≥5, hence BO(3, Z11) = 5;

– the set {0,1,3,9}shows that BO(3, Z13 )≥5 and, by Theorem 3.8 we

have BO(3, Z13)≤6,upper bound improved in the next result.

Lemma 3.9.BO(3, Z13) = 5.

Proof. The action of group G13 on X3

13 partitions it into three orbits:

θ({0,1,2}) (barycentric) and θ({0,1,3}), θ({0,1,4)}) (non barycentric). Now,

we apply Remark 2.4 to the orbits θ({0,1,3}), θ({0,1,4)}) in order to obtain 3-

barycentric-free orbits in X4

13. These orbits are: θ({0,1,3,4}), θ({0,1,3,9}),

θ({0,1,3,11}), θ({0,1,4,5}), θ({0,1,4,6}), θ({0,1,4,10}). Finally, none of

these orbits can be extended by a ﬁfth element without forming a 3-barycentric

subset.

Lemma 3.10.BO(3, Zn) = 5 for n= 6,8,9,10.

Proof. n= 6. Under the action of group G6, the set X3

6has the follow-

ing partition by orbits: θ({0,1,2}), θ({0,2,4}) (barycentric) and θ({0,1,3})

(non barycentric). The only 3-barycentric-free orbit in X4

6obtained from

θ({0,1,3}), is θ({0,1,3,4}), thus BO(3, Z6)>4. The only orbit in X5

6ob-

tained from θ({0,1,3,4}) contains a 3-barycentric set. Then BO(3, Z6)≤5.

n= 8. Under the action of group G8, the set X3

8has the following parti-

tion by orbits: θ({0,1,2}), θ({0,2,4}) (barycentric) and θ({0,1,3}), θ({0,1,4})

(non barycentric). The only 3-barycentric-free orbits in X4

8obtained from

the non barycentric orbits in X3

8, are θ({0,1,3,4}) and θ({0,1,4,5}), thus

BO(3, Z8)>4. None of these orbits can be extended by a ﬁfth element in

order to form 3-barycentric-free orbits in X5

8. Then BO(3, Z8)≤5.

n= 9. Under the action of group G9, the set X3

9has the following

partition by orbits: θ({0,1,2}), θ({0,3,6}) (barycentric) and θ({0,1,6}) (non

barycentric). The only 3-barycentric-free orbits in X4

9obtained from θ({0,1,6}),

are θ({0,1,3,4}) and θ({0,1,3,7}). Thus BO(3, Z9)>4. None of these orbits

can be extended by a ﬁfth element in order to form 3-barycentric-free orbits

in X5

9. Then BO(3, Z9)≤5.

7k-barycentric Olson constant 39

n= 10. Under the action of group G10, the set X3

10 has the follow-

ing partition by orbits: θ({0,1,2}), θ({0,2,4}) (barycentric) and θ({0,1,3}),

θ({0,1,5}) (non barycentric). The only 3-barycentric-free orbits in X4

10 ob-

tained from the non barycentric orbits in X3

10 are θ({0,1,3,4}), θ({0,1,3,8}),

θ({0,1,4,5}) and θ({0,1,5,6}). Thus BO(3, Z10 )>4. None of these orbits

can be extended by a ﬁfth element in order to form 3-barycentric-free orbits

in X5

10. Then BO(3, Z10 )≤5.

Lemma 3.11.BO(3, Z4) = 3.

Proof. Lemma 2.6.

Lemma 3.12.BO(4, Z7) = 5.

Proof. Under the action of G7, the only two orbits in X4

7are θ({0,1,2,3})

(non barycentric, thus 5 ≤BO(4, Z7)) and θ({0,1,2,4}) (barycentric). By

a simple inspection we can see that the only orbit in X5

7, obtained from

θ({0,1,2,3}), contains a 4-barycentric set. Then BO(4, Z7)≤5. By Lem-

mas 2.3 and 2.6, X5

7contains only one orbit, say θ({0,1,2,3,4}).Moreover,

this orbit contains a 4-barycentric set. Therefore, this is another way to obtain

the upper bound BO(4, Z7)≤5.

The two theorems below and their corollaries give other bounds for

BO(k, Zp).

Theorem 3.13 ([6]).Let s, d be integers ≥2such that s > lp−1

dm.

Let Abe a set with |A|=s+d. Let f:A→Zpbe a sequence such that

|f(A)| ≥ lp−1

dm+d. Then there exists an s-barycentric subsequence of f.

Corollary 3.14.BOlp−1

dm+ 1, Zp≤lp−1

dm+ 1 + dfor d≥2and

p≥lp−1

dm+1+d.

Theorem 3.15 ([6]).Let s≥2,p≥7and Aa set with s+2 elements. If

f:A→Zpis a sequence with |f(A)|=p+3

2, then fcontains an s-barycentric

subsequence.

Corollary 3.16.BOlp−1

2m, Zp≤p+3

2for p≥7.

In particular, we have BO(5, Z11)≤7.

We have the following easy

Lemma 3.17.θ({0,1,2})is a barycentric orbit of X3

n,n≥3.

θ({0,1,2,5})is a barycentric orbit of X4

n,n≥6.

θ({0,1,2,3,4})is a barycentric orbit of X5

n,n≥5.

40 Oscar Ordaz, Mar´ıa Teresa Varela and Felicia Villarroel 8

θ({0,1,2,3,5,7})is a barycentric orbit of X6

n,n≥8.

θ({0,1,2,3,4,5,6})is a barycentric orbit of X7

n,n≥7.

θ({0,1,2,3,4,5,6,7})is a barycentric orbit of X8

n,n= 10,12.

θ({0,1,2,3,4,7,8,9})is a barycentric orbit of X8

11.

θ({0,1,2,3,4,5,6,7,8})is a barycentric orbit of X9

n,n≥9.

θ({0,1,2,3,4,5,6,8,10,11})is a barycentric orbit of X10

12 .

θ({0,1,2,3,4,5,6,7,8,9,10})is a barycentric orbit of X11

n,n≥11.

Remark 3.18.The existence of BO(k, Zn) for 3 ≤n≤12 and 3 ≤k≤n

is ensured by Remark 3.2 and Lemma 3.17.

The exact values of the k-barycentric Olson constant are presented in

Table 1 and Table 2. The values that are not justiﬁed by lemmas are obtained

similarly using orbits technique as in Lemmas 3.9 through 3.12. In the lower

bound column, a longest set without a k-barycentric set is given.

TABLE 1

Exact values of BO(k, G)

Lower bound Justiﬁcation k G BO(k , G)

{0,1}3Z33

{0,1}Lemma 3.11 3 Z43

{0,1}3Z53

{0,1,3,4}Lemma 3.10 3 Z65

{0,1,3}3Z74

{0,1,3,4}Lemma 3.10 3 Z85

{0,1,3,4}3Z95

{0,1,3,4}Lemma 3.10 3 Z10 5

{0,1,4,5}3Z11 5

{0,1,3,4}3Z12 5

Remark 3.2 4 Z4non deﬁned

Remark 3.2 4 Z5non deﬁned

{0,1,3,4}4Z65

{0,1,2,3}Lemma 3.12 4 Z75

{0,1,2,5,7}4Z86

{0,1,3,4,6,7}4Z97

{0,1,2,3,4}4Z10 6

{0,1,2,3,4}4Z11 6

{0,1,2,3,4}4Z12 6

Remark 3.2 5 Z55

{0,1,2,3}5Z65

{0,1,2,4}5Z75

{0,1,2,3,5}5Z86

{0,1,2,3,5}5Z96

{0,1,2,3,5,8}5Z10 7

{0,1,2,6,8,9}5Z11 7

{0,1,2,3,5,7}5Z12 7

9k-barycentric Olson constant 41

TABLE 2

Exact values of BO(k, G)

Lower bound Justiﬁcation k G BO(k, G)

Remark 3.2 6 Z6non deﬁned

Remark 3.2 6 Z7non deﬁned

{0,1,2,3,4,5}6Z87

{0,1,2,3,4,6}6Z98

{0,1,2,3,4,5}6Z10 7

{0,1,2,3,5,10}6Z11 7

{0,1,2,3,5,8}6Z12 7

{0,1,2,3,4,5}Remark 3.2 7 Z77

{0,1,2,3,4,5}7Z87

{0,1,2,3,4,5}7Z97

{0,1,2,3,4,5,7}7Z10 8

{0,1,2,3,5,7,10}7Z11 8

{0,1,3,4,5,7,10}7Z12 8

Remark 3.2 8 Z8non deﬁned

Remark 3.2 8 Z9non deﬁned

{0,1,2,3,4,5,6,8}8Z10 9

{0,1,2,3,5,7,9,10}8Z11 9

{0,1,2,3,4,5,7,8}8Z12 9

{0,1,2,3,4,6,7,8}Remark 3.2 9 Z99

{0,1,2,3,5,6,7,8}9Z10 9

{0,1,2,3,4,5,6,7}9Z11 9

{0,1,2,3,4,5,6,7}9Z12 9

Remark 3.2 10 Z10 non deﬁned

Remark 3.2 10 Z11 non deﬁned

{0,1,2,3,4,5,6,7,8,9}10 Z12 11

{0,1,2,3,4,5,6,7,8,10}Remark 3.2 11 Z11 11

{0,1,2,3,4,5,6,7,8,9}11 Z12 11

Remark 3.2 12 Z12 non deﬁned

4. k-BARYCENTRIC DAVENPORT CONSTANT

AND BARYCENTRIC RAMSEY NUMBERS FOR STARS

Let Gbe an Abelian group of order nand k≤n. If BO(k, G) exists,

then we have BO(k, G)≤BD(k, G), otherwise n+ 1 ≤BD(k, G).From the

Hamidoune-condition we have BD(k, G)≤n+k−1, i.e., so that its existence

is assured. When k=|G|=n, the k-barycentric sequences are zero-sum.

Gao [9] showed that ZS(G) = n+D(G)−1, where ZS(G) is the smallest

positive integer tsuch that every sequence of length tcontains a zero-sum

n-subsequence. Then B D(n, G) = Z S(G).Moreover, since D(Zn) = n(see

[17]), we have BD(n, G) = 2n−1.

42 Oscar Ordaz, Mar´ıa Teresa Varela and Felicia Villarroel 10

New values of BD(k, G) diﬀerent from those presented in [7] are given

here.

In this section, the values of BR(K1,k, Zn) for 3 ≤n≤12 and 3 ≤k≤n

are established. The upper bound is derived from the fact that BR(K1,k, Zn)≤

BD(k, Zn) + 1. The process of obtaining the lower bound is as follows. Let µ

be the sequence that yields the lower bound of BD(k, Zn).Select an adequate

decomposition of the complete graph KBD(k,Zn), using Theorem 2.9, Corol-

lary 2.10 and Corollary 2.11. The edges of each subgraph, part of this decom-

position, are colored; µis the sequence of edge colors of the K1,B D(k,Zn)−1stars.

In some cases, such as BR(K1,4, Z6) = 7, its upper bound is derived from

BO(4, Z6) = 5: if there exists a K1,6in K7colored with ﬁve or six colors, we

are done. Else, if all K1,6are colored with three or four colors, then each orbit

of X3

6,X4

6, respectively, is used to color the edges of K7. For example, let

θ({0,1,2}) be an orbit of X3

6. It is easy to see that coloring any K1,6in K7

with 0,1,2 we can identify a K1,4barycentric. If all K1,6are colored with only

two colors; then they must be colored by {0,1},{0,2}or {0,3}. Therefore,

assuming that K7is K1,4barycentric free, the graph induced by the edges

colored by 0 has odd order and each vertex has odd degree, and we reach

a contradiction. For the lower bound, Theorem 2.9 is used. The two-edge

disjoint Hamiltonian cycles of K6are colored by 1 and 3, respectively, and the

perfect matching by 0.

In what follows we give a few representative examples to illustrate the

methods used to compute BD(k, G).

Lemma 4.1.BD(3, Z4) = 5.

Proof. Since BO(3, Z4) = 3, the sequence that yields the lower bound

must have at most two diﬀerent elements. Since the orbits of X2

4are θ({0,1})

and θ({0,2}), the length of 3-barycentric free sequence 0011 is a lower bound

for BD(3, Z4).Moreover, in every sequence of length 5, there exist at least

three diﬀerent elements or three identical elements. Then it contains a 3-

barycentric sequence. The proof is complete.

Lemma 4.2.BD(3, Z6) = 5.

Proof. Since BO(3, Z6) = 5, the lower bound is obtained for sequences

with four diﬀerent elements. For example, the sequence 0134 is 3-barycentric

free. Moreover, if in a sequence of length 5 there exists an element repeated

three times, then we are done. Otherwise, it contains three diﬀerent elements.

Since the only orbit in X3

6not barycentric is θ({0,1,3}), it is suﬃcient to con-

sider the sequence 00113, which contains the 3-barycentric sequence 003.

11 k-barycentric Olson constant 43

In Table 3 and Table 4 we give the exact values of BD(k, Zn) and

BR(K1,k , Zn) for 3 ≤n≤12 and 3 ≤k≤n. In the lower bound column

is given a longest sequence without a k-barycentric subsequence.

TABLE 3

Exact values of BD(k, G) and BR(K1,k, G)

Lower bound k G BD(k, G)B R(K1,k , G)

0101 3 Z35 6

0101 3 Z45 6

0101 3 Z55 6

0113 3 Z65 6

013013 3 Z77 8

013013 3 Z87 8

01340134 3 Z99 10

01340134 3 Z10 9 10

01340134 3 Z11 9 10

01340134 3 Z12 9 10

000111 4 Z47 7

000111 4 Z57 8

000111 4 Z67 7

0001222 4 Z78 9

01450011 4 Z88 9

0120022 4 Z98 9

01560011 4 Z10 9 10

01341133 4 Z11 9 10

01270022 4 Z12 9 10

00001111 5 Z59 10

00001111 5 Z69 10

00001111 5 Z79 10

00001111 5 Z89 10

00001111 5 Z99 10

00001111 5 Z10 9 10

00001111 5 Z11 9 10

000011115 5 Z12 10 11

5. CONCLUSION

The k-barycentric Olson constants BO(k, G) play an important role in

the computation of other constants such as the k-barycentric Davenport con-

stant BD(k, G) and the barycentric Ramsey numbers stars BR(K1,k , G). We

only discussed the existence of BO(k, G) for some group G. It is an open prob-

lem to prove the existence of and to compute BO(k, G) for a general Abelian

group Gby using results of additive group theory.

44 Oscar Ordaz, Mar´ıa Teresa Varela and Felicia Villarroel 12

TABLE 4

Exact values of BD(k, G) and BR(K1,k, G)

Lower bound k G BD(k , G)BR(K1,k, G)

0000011111 6 Z611 11

0000011111 6 Z711 12

0000011111 6 Z811 11

0000011111 6 Z911 12

0000011111 6 Z10 11 11

00000122222 6 Z11 12 13

000001111167 6 Z12 13 14

000000111111 7 Z713 14

000000111111 7 Z813 14

000000111111 7 Z913 14

000000111111 7 Z10 13 14

000000111111 7 Z11 13 14

000000111111 7 Z12 13 14

00000001111111 8 Z815 15

00000001111111 8 Z915 16

00000001111111 8 Z10 15 15

00000001111111 8 Z11 15 16

00000001111111 8 Z12 15 15

0000000011111111 9 Z917 18

0000000011111111 9 Z10 17 18

0000000011111111 9 Z11 17 18

0000000011111111 9 Z12 17 18

000000000111111111 10 Z10 19 19

000000000111111111 10 Z11 19 20

000000000111111111 10 Z12 19 19

00000000001111111111 11 Z11 21 22

00000000001111111111 11 Z12 21 22

0000000000011111111111 12 Z12 23 23

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Received 2 November 2007 Universidad Central de Venezuela

Facultad de Ciencias

Departamento de Matem´aticas y Centro ISYS

Ap. 47567, Caracas 1041-A, Venezuela

ﬂosav@cantv.net

Universidad Sim´on Bolivar

Departamento de Matem´aticas Puras y Aplicadas

Ap. 89000, Caracas 1080-A, Venezuela

and

Universidad de Oriente

Departamento de Matem´aticas

Escuela de Ciencias, N´ucleo Sucre

Cuman´a, Venezuela