\documentclass[12pt]{article} \textwidth= 6.25in \textheight= 9.0in \topmargin = -10pt \evensidemargin=10pt \oddsidemargin=10pt \headsep=25pt \parskip=10pt \font\smallit=cmti10 \font\smalltt=cmtt10 \font\smallrm=cmr9 \usepackage{amsmath, amssymb,amsthm} \def\z{{\bf Z}} \def\zn{{\bf Z}_n} \def\zm{{\bf Z}_m} \def\zp{{\bf Z}_p} \def\zpo{{\bf Z}_p\oplus{\bf Z}_p} \def\zno{{\bf Z}_n\oplus{\bf Z}_n} \def\zmo{{\bf Z}_m\oplus{\bf Z}_m} \begin{document} \begin{center} {\large\bf{NON-CANONICAL EXTENSIONS OF ERD\H{O}S-GINZBURG-ZIV THEOREM}\footnote{Mathematics subject classification (1991): 05D05, 20D10 } } \vskip 20pt {\bf R. Thangadurai}\\ {\smallit Stat-Math Division, Indian Statistical Institute, 203, B. T. Road, Kolkata 700035, INDIA}\\ {\tt thanga\_v@isical.ac.in} \end{center} \vskip 30pt \centerline{\smallit Received: 11/28/01, Revised: 4/2/02, Accepted: 4/8/02, Published: 4/15/02} \vskip 30pt \centerline{\bf Abstract} \noindent In 1961, Erd\H{o}s-Ginzburg-Ziv proved that for a given natural number $n\geq 1$ and a sequence $a_1,a_2,\cdots,a_{2n-1}$ of integers (not necessarily distinct), there exist $1\leq i_1 < i_2 < \cdots < i_n \leq 2n-1$ such that $a_{i_1}+a_{i_2}+\cdots+a_{i_n}$ is divisible by $n.$ Moreover, the constant $2n-1$ is tight. By now, there are many canonical generalizations of this theorem. In this paper, we shall prove some non-canonical generalizations of this theorem. \pagestyle{myheadings} \markright{\smalltt INTEGERS: \smallrm ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 2 (2002), \#A07\hfill} \thispagestyle{empty} \baselineskip=15pt \vskip 30pt \section*{\normalsize 1. Introduction and Preliminaries} \addtocounter{section}{+1} Additive number theory, graph theory and factorization theory provide inexhaustible sources for combinatorial problems in finite abelian groups (cf. \cite{man1}, \cite{man2}, \cite{eg}, \cite{dm}, \cite{na} and \cite{an}). Among them zero sum problems have been of growing interest. Starting points of recent research in this area were the Theorem of Erd\H{o}s-Ginzburg-Ziv (EGZ Theorem, for short) and a question of H. Davenport on an invariant which today carries his name. We shall denote the cyclic group of order $n$ by $\zn.$ A sequence $S = \{a_i\}_{i=1}^\ell$ of length $\ell$ in $\zn,$ we mean $a_i \in \zn$ and $a_i$'s are not necessarily distinct, unless otherwise specified. Also, throughout this paper, writting $n \geq 1,$ we mean $n$ is an arbitrary natural number and writting $p,$ we mean an arbitrary prime number. We shall define some terminalogies as follows. A sequence $S$ is called {\sf zero sequence} if its sum is zero. A sequence $S$ is called {\sf zero-free sequence} if it contains no zero subsequence. A sequence $S$ is called {\sf minimal zero sequence} if $S$ is a zero sequence; but any proper subsequence is zero-free. Now, we shall restate the EGZ theorem, using the above terminalogies, as follows. \bigskip \noindent{\bf EGZ Theorem.} (cf. \cite{egz}) {\it Given a sequence $S$ in $\zn$ of length $2n-1,$ one can extract a zero subsequence of length $n$ in $\zn.$} \bigskip We shall state the following known result which will be useful for our further discussion. \bigskip \noindent{\bf Cauchy-Davenport inequality.} {\it Let $A$ and $B$ be two non-empty subsets of $\zp.$ Then $$ |A+B| \geq \min\{p, |A| + |B| -1\} $$ where $A+B = \{x = a+b \in \zp : a, b \in \zp\}$ and $|K|$ denotes the cardinality of the subset $K$ of $\zp.$} \smallskip This was first proved by Cauchy (cf. \cite{cu1}) in 1813 and was rediscovered by Davenport (cf. \cite{da1}) in 1947. \bigskip \noindent{\bf Corollary 1.1} {\it Let $A_1, A_2, \cdots, A_h$ be non-empty subsets of $\zp.$ Then, $$ |A_1+A_2+\cdots+A_h | \geq \min\{p, \sum_{i=1}^h|A_i| - h + 1\}. $$} By today there are several extensions of Erd\H{o}s-Ginzburg-Ziv theorem are known. All the known extensions are natural and we call them canonical extensions. In this paper, we shall prove several non-canonical extensions of EGZ theorem. \section*{\normalsize 2. Canonical extensions of EGZ Theorem} \addtocounter{section}{+1} In this section, we shall survey the results which are natural generalization or extensions of EGZ Theorem and we call them as C-Extensions. The first natural generalization of EGZ in $\bf Z$ is the following due to Olson. \bigskip \noindent{\bf C-Extension 1.} (Olson, 1969, \cite{o1}) {\it Suppose $m \geq k \geq 2$ are integers such that $k|m.$ Let $a_1, a_2,$ $\cdots, a_{m+k-1}$ be a sequence of integers. Then there exists a non-empty subset $I$ of $\{1,2,\cdots,m+k-1\},$ such that $|I| = m$ and $\sum_{i\in I}a_i\equiv 0 \pmod{k}.$} \bigskip If one view EGZ theorem as a statement over the solvable group $\zn,$ then one can ask for the same in any finite group. Indeed, Olson proved that \bigskip \noindent{\bf C-Extension 2.} (Olson, 1976, \cite{o2}) {\it Let $g_1,g_2, \cdots,g_{2n-1}$ be a sequence of length $2n-1$ in a finite group (but not necessarily Abelian) $G$ of order $n.$ Then there exists elements $g_{i_1}, g_{i_2}, \cdots, g_{i_n}$ from the given sequence satisfies $g_{i_1}+g_{i_2}+ \cdots+g_{i_n} = 0$ in $G.$} \bigskip \noindent{\bf Conjecture 2.1} (Olson, 1976, \cite{o2}) {\it The same conclusion holds in C-Extension 2, together with $i_1 < i_2 < \cdots < i_n.$} \bigskip In this direction, W. D. Gao (cf. \cite{ga1}) proved in 1996 that if $G$ is a non-cyclic solvable group and $s = [11n/6]-1,$ then for any sequence $a_1,a_2,\cdots, a_s$ in $G,$ we have $i_1, i_2, \cdots, i_n$ distinct such that $a_{i_1}+ \cdots+ a_{i_n} = 0 $ in $G.$ Other than this result, we know nothing about Conjecture 2.1. In C-Extension 2, $2n-1$ may not be tight except for the group $G \cong \zn.$ Indeed, if $G$ is an abliean group (additively written) of order $n,$ then Gao (cf. \cite{ga5}) proved that $n+D(G)-1$ is the right constant in place of $2n-1$ where $D(G)$ (is the {\sf Davenport Constant}), which is the least positive integer such that given any sequence $S$ in $G$ of length $\ell(S)$ with $\ell(S) \geq D(G),$ there exists a zero subsequence $T$ of $S$ in $G.$ One can easily see that when $G \cong \zn,$ we have $D(\zn) = n.$ \bigskip There is yet another generalization of EGZ theorem as follows. EGZ theorem proves the existence of one zero subsequence of length $n,$ whenever we consider a sequence in $\zn$ of length $2n-1.$ \bigskip \noindent{\bf C-Extension 3} (W. D. Gao, 1997, \cite{ga7}) {\it If $a_1,a_2, \cdots,a_{2n-1}$ be a sequence in $\zn,$ then there exists at least $n$ number of subsequences of length $n$ having its sum `$a$' for any given $a \in {\bf Z}_n^*$ provided no element occurs more than $n$ times in the sequence. More over, there exists at least $n+1$ number of zero subsequences of length $n$ in $\zn$ unless only two elements $x$ and $y$ occur $n$ and $n-1$ times respectively in that sequence.} \bigskip Indeed, chronologically, H. B. Mann (cf. \cite{ma1}) in 1967 proved the existence of one subsequence of length $p$ whose sum is $g$ for a given $g \in \zp,$ whenever we consider a sequence in $\zp$ of length $2p-1.$ In 1996, W. D. Gao (cf. \cite{ga2}) proved that C-Extension 3 for $n = p$ prime. (Indeed, Sury (cf. \cite{su1}) gave a different proof of this fact). \bigskip One should mention the result of Bialostocki and Dierker which is a stronger version of EGZ theorem; \noindent{\bf C-Extension 4'.} (Bialostocki and Dierker, 1992, \cite{bd1}) {\it If $S = \{a_i\}$ is a sequence in $\zp$ of length $2p-1,$ then there are $p$ indices $1\leq i_i < i_2 < \cdots < i_p \leq 2p-1$ such that $$a_{i_1} + a_{i_2}+\cdots+a_{i_p} \equiv 0\pmod{p}.$$ Moreover, if for two indices $j, k$ we have $a_j \not\equiv a_k\pmod{p},$ then we can choose $i_1, i_2, \cdots, i_p$ such that not both $j$ and $k$ are among them in that zero of length $p.$} \bigskip Indeed, we shall prove better result than C-Extension 4' as follows. \bigskip \noindent{\bf C-Extension 4.} {\it If $S = \{a_i\}$ is a sequence in $\zp$ of length $2p-1,$ then there are $p$ indices $1\leq i_i < i_2 < \cdots < i_p \leq 2p-1$ such that \begin{eqnarray} a_{i_1} + a_{i_2}+\cdots+a_{i_p} \equiv 0\pmod{p}. \end{eqnarray} Moreover, if $s \geq 2$ distinct elements of $\zp,$ say, $a_1,a_2,\cdots,a_s$ are in $S,$ then we can choose the $i_1, i_2, \cdots, i_p$ such that only one of the indices from $1,2,\cdots,s$ appears among $i_1, i_2, \cdots, i_p$ and satisfying $(1).$} \smallskip \noindent{\bf Proof.} If one of the element of $S$ is repeated more than $p$ times, then the result follows trivially. Assume that none of the elements of $S$ appears more than $p-1$ times. Let $A_1 = \{a_1,a_2,\cdots,a_s\} \subset \zp$ and the remaining $a_i$'s be distributed into non-empty $p-1$ incongruent classes modulo $p,$ say, $A_2, A_3,\cdots, A_{p-1}.$ Then by Cauchy-Davenport inequality, we have $$|A_1+A_2+\cdots+A_p| \geq \min\{p, \sum_i |A_i| -p +1\} = \min\{p,2p-1-p+1\} = p$$ $$\Longrightarrow A_1+A_2+\cdots +A_p = \zp.$$ Thus the result follows. $\hfill\Box$ \section*{\normalsize 3. Non-Canonical extensions of EGZ Theorem} \addtocounter{section}{+1} In this section, we prove non-canonical generalizations of EGZ Theorem and we call them as N-C Extensions. \bigskip \noindent{\bf N-C Extension 1.} {\it Let $S$ be a sequence in $\zn$ of length at least $n.$ Let $h = h(S) = \max_{a \in S} g(a)$ where $g(a)$ denote the number of times $a \in \zn$ appearing in $S.$ Then there is a zero subsequence of length less than or equal to $h.$ } \bigskip This theorem was proved by Gao and Yang (cf. \cite{gy1}) in 1997. We shall prove that this theorem indeed implies EGZ Theorem. \bigskip \noindent{\bf Theorem 3.1} {\it N-C Extension 1 implies EGZ theorem.} \bigskip Before going to the proof of Theorem 3.1, we prove the following lemma. \bigskip \noindent{\bf Lemma 3.1.1} {\it Let $S$ be a sequence in $\zn$ of length $2n-1.$ Suppose there is an element $a\in\zn$ such that $a$ is appearing in $S$ at least $[n/2]$ times, then there is a zero subsequence of $S$ of length $n$ in $\zn.$} \smallskip \noindent{\bf Proof.} Let $S$ be a sequence in $\zn$ of length $2n-1.$ Suppose $S$ consists of an element $a \in \zn$ which is repeated $s \geq [n/2]$ number of times. If $s\geq n,$ then the result is obvious. Let us assume that $s \leq n-1.$ Consider the translated sequence $S-a$ in which $0$ is repeated $s$ number of times. The length of the subsequence $T_1$ of $S-a$ which consists of all the non-zero elements of $S-a$ is $2n-1 -s \geq n.$ Since $D(\zn) = n,$ the sequence $T_1$ contains a zero subsequence say $T_2.$ Let the length of $T_2$ be $t_2.$ Clearly, $2 \leq t_2 \leq n.$ Choose $T_2$ such that it has the maximal length $t_2.$ Also, note that if $s+t_2 \geq n,$ then we can extract a zero of length $n$ in $S-a$ which in turn produces a zero subsequence of length $n$ in $S.$ Thus we can assume that $s+t_2 < n.$ Note that $[n/2]+1 \leq t_2\leq n.$ If not, that is, $t_2 \leq [n/2].$ Then after omitting $T_2$ from $T_1,$ the length of the sequence $T_1 \backslash T_2$ is at least $n$ and hence it contains a zero subsequence say $T_3$ with length $t_3.$ Clearly, $t_3 \leq t_2 \leq [n/2],$ which implies, $t_2+t_3 \leq n,$ which contradicts to the maximality of $T_2.$ Hence $[n/2]+1 \leq t_2 \leq n$ is true. Since we have at least $[n/2]$ zeros out side $T_2,$ by adding appropriate number of zeros to $T_2,$ we get a zero sequence of length $n$ in $S-a$ which in turn produces a zero sequence of length $n$ in $S.$ $\hfill\Box$ \bigskip \noindent{\bf Proof of Theorem 3.1.} Let $S$ be a sequence in $\zn$ of length $2n-1.$ Suppose $S$ consists of an element $a \in \zn$ which is repeated maximum number of $s$ times. If $s \geq n,$ then nothing to prove. \smallskip \noindent{\bf\sc Case (i)} $([n/2] \leq s \leq n-1)$ \smallskip This case is covered by Lemma 3.1.1. \smallskip \noindent{\bf\sc Case (ii)} $(2\leq s \leq [n/2]-1)$ \smallskip Consider the translated sequence $S-a$ inwhich $0$ is repeated $s$ times. The length of the subsequence $T_1$ of $S-a$ which consists of all the non-zero elements of $S-a$ is $2n-1 -s \geq n+n-[n/2].$ Since $D(\zn) = n,$ there exists a zero subsequence of $T_2$ of length $t.$ Choose $T_2$ having the maximum length. Then, it follows that (apply the same argument given in the begining of the proof of Lemma 3.1) $[n/2]+1 \leq t\leq n.$ \smallskip \noindent{\bf\sc Claim.} $s+t > n.$ \smallskip Suppose not, that is, $s+t < n.$ Now delete the subsequence $T_2$ from $T_1.$ Then the length of the deleted sequence, say $T_3,$ is $2n-1-s-t \geq 2n-1-(n-1) = n+1.$ By N-C Extension 1, there exists a zero subsequence of length less than or equal to $s$ in $T_3.$ Therefore there exists a subsequence $T_4$ of $T_3$ such that the length of $T_4,$ say $t_1$ is less than or equal to $s.$ Since $T_2$ is maximal with respect to its length less than or equal to $n,$ it is clear that $t+t_1 > n.$ If $2 \leq t_1\leq s\leq [n/2]-1,$ then $n-t_1+1 \leq t \leq n-1.$ Since $t_1\leq s,$ $n-t_1+1 \geq n-s+1$ which implies that $s+t > n$ which is a contradiction to the assumption. This proves the claim. Since $s+t > n$ and $t \leq n,$ we can add appropriate number of zeros to $T_2$ so as to get a zero sequence of length $n.\hfill\Box$ \bigskip Before going into the further discussions, we shall prove the following theorem. \bigskip \noindent{\bf Theorem 3.2.} {\it Let $n$ and $k$ be positive integers such that $1 \leq k < (n+2)/3.$ Then, the following statments are equivalent. (I) Let $S$ be a minimal zero sequence of $\zn$ of length $n-k+1.$ Then there exists $a\in\zn$ such that $a$ appears in $S$ at least $n-2k+2$ times. (II) Let $S$ be a zero-free sequence in $\zn$ of length $n-k.$ Then one element of $S$ is repeated at least $n-2k+1$ times. (III) Let $S$ be a sequence of $\zn$ of length $2n-k-1.$ Suppose $S$ does not have a zero subsequence of length $n.$ Then there exist $a \ne b \in \zn$ such that $a$ and $b$ appear in $S$ at least $n-2k+1$ times.} \bigskip \noindent{\it Remark.} The statement (I) was proved by the author in 2001 (cf. \cite{th1}). The statement (II) was proved by Bovey, Erd\H{o}s and Niven in 1975 (cf. \cite{ben1}). Also, all the three statements are valid for $n-2k \geq 1.$ But the equivalence is valid only for the range $1 \leq k < (n+2)/3.$ \bigskip \noindent{\bf Proof.} {\sf (I) $\iff$ (II)} Assume that (I) is true. Consider a zero-free sequence $S = \{a_i\}$ in $\zn$ of length $n-k.$ Let $a_{n-k+1} = -\sum_{i=1}^{n-k}a_i$ and $S_1$ be the sequence consisting of all the elements $a_i$ together with $a_{n-k+1}.$ Then, $S_1$ is a minimal sequence of length $n-k+1$ in $\zn.$ For, if any proper subsequence, say, $T$ of $S$ together with $a_{n-k+1}$ is a zero subsequence of $S_1,$ then the deleted sequence $S\backslash T$ is a zero subsequence of $S$ which is a contradiction. Hence $S_1$ is a minimal zero sequence of length $n-k+1$ in $\zn.$ Now, by the statement (I), there exists $a \in \zn$ such that $a$ is repeated in $S_1$ at least $n-2k+2$ times. Thus, the element $a$ is repeated in $S$ at least $n-2k+1$ times. Assume that (II) is true. Consider a minimal zero sequence $S = \{a_i\}$ in $\zn$ of length $n-k+1.$ Let $S_1$ be the sequence obtained from $S$ by deleting the element $a_{n-k+1}.$ Clearly, $S_1$ is a zero-free sequence in $\zn$ of length $n-k.$ Therefore, by the statement (II), there exists $a \in \zn$ such that $a$ is repeated in $S_1$ at least $n-2k+1$ times. Now, let $S_2$ be a zero sequence in $\zn$ of length $n-k,$ obtained from $S$ by deleting the element $a\ne a_{n-k+1}$ (if $a = a_{n-k+1},$ then nothing to prove). Again by the statement (II), there exists an element $b\in \zn$ such that $b$ is repeated $n-2k+1$ times in $S_2.$ If $a \ne b,$ then the length of $S$ would be at least $2n -4k+2 \leq n-2k +1.$ This forces that $n \leq 2k -1.$ That is, $k \geq (n+1)/2,$ which is a contradition to the assumption that $k \leq (n+2)/3.$ Therefore, $a = b$ and hence $S$ has an element $a\in\zn$ which is repeated in $S$ at least $n-2k+2$ times. \bigskip \noindent{\sf (II) $\iff$ (III)} \bigskip Assume that (II) is true. Let $S$ be a sequence in $\zn$ of length $2n-k-1$ satisfying the hypothesis. Suppose $a$ is an element of $S$ which appears maximum number of, say $h$, times in $S.$ Consider the translated sequence $S -a.$ Let $S_1$ be the subsequence of $S-a$ such that it consists of all non-zero elements of $S-a.$ \smallskip \noindent{\bf\sc Claim.} There exists zero-free subsequence $T$ of $S_1$ of length $n-k.$ \smallskip Assume the contrary. Suppose every subsequence $T$ of $S_1$ of length $n-k$ has a zero subsequence. Let $M$ be one such zero subsequence of $S_1.$ Choose $M$ such that $M$ has the maximal length. Since every subsequence of length $n-k$ of $S_1$ has a zero subsequence, it is clear that $$ 2n-k-1-h-|M| \leq n-k-1 \Longrightarrow |M| \geq n-h. $$ If $|M| \leq n,$ then by adding appropriate number of zeros to $M$ to get a zero subsequence of length $n$ in $S-a$ which in turn produces a zero subsequence of length $n$ in $S$ (this is because we have $h$ zeros out side $S_1$). This is a contradiction to the assumption. Therefore, $|M| > n.$ In this case, by N-C Extension 1, we can get a zero subsequence of length $\leq h,$ in $M.$ Therefore, inductively, deleting the zero subsequences $M_1, M_2, \cdots, M_r$ from $M$ so that we can make $n-h \leq |M'| \leq n$ where $M'$ is the sequence obtained from $M$ after deleting those sequences $M_i$'s. This is a contradiction as before. This contradiction implies that there is a zero-free subsequence $T$ of $S_1$ of length $n-k.$ Therefore, by the statement (II), $T$ has an element which is repeated at least $n-2k+1$ times. Since $0$ appears in $S-a$ maximum number of times, $h \geq n-2k+1.$ Therefore two distinct elements of $\zn$ in $S-a$ which appears at least $n-2k+1$ times. Hence $S$ has two distinct elements of $\zn$ such that both appears in $S$ at least $n-2k+1$ times. Assume that (III) is true. Let $S$ be a zero-free sequence of $\zn$ of length $n-k.$ Let $$S_1 : S, {\underbrace{0,0,\cdots,0}_{n-1 \ {\rm times}}}$$ be sequence in $\zn$ of length $2n-k-1.$ Clearly $S_1$ does not contain a zero subsequence of length $n.$ Therefore by the statement (III), we know, $S_1$ consists of two distinct elements of $\zn$ such that both appears $n-2k+1$ times. Since $0$ appears $n-1$ times, it is clear that $S$ consists of one element of $\zn$ which is repeated at least $n-2k +1$ times. $\hfill\Box$ \bigskip \noindent{\it Remark 1.} The statements (II) and (III) are equivalent for all $n$ and $k$ such that $n -2k \geq 1.$ Also, in the statement (III), there is a moreover part. That is, we can prove that, in the conclusion of the statement (III), $S$ can consists of at most $k+1$ distinct residue classes modulo $n.$ This is because of the following. In \cite{bl1}, it is proved that if any sequence $R$ in $\zn$ of length $2n-m+1$ consists of $m$ distinct residue classes modulo $n,$ then $R$ contain a zero subsequence of length $n.$ Since the length of the given sequence $S$ is $2n-k-1$ and $S$ doesn't have any zero subsequence of length $n,$ it follows that $S$ can contain at most $k+1$ distinct residue classes modulo $n.$ \bigskip \noindent{\bf Theorem 3.3} {\it Let $k$ be an integer with $1 \leq k \leq \frac{1}{2}(n-[n/2]+1).$ Then, the statement (III) in Theorem 3.2 implies EGZ Theorem.} \smallskip \noindent{\bf Proof.} Let $S$ be a sequence in $\zn$ of length $2n-1.$ Let $T$ be a subsequence of length $2n-1-k$ where $1 \leq k \leq \frac{1}{2}(n-[n/2]+1).$ Either $T$ has a zero subsequence of length $n$ or it doesn't have. If $T$ has such a zero subsequence, then nothing to prove. If $T$ doesn't have any zero subsequence of length $n,$ then by the statement (III) in Theorem 3.2, $T$ consists of two distinct elements of $\zn$ each appearing $n-2k+1$ times. Since $k$ lies in $1 \leq k \leq \frac{1}{2}(n-[n/2]+1),$ we get $n-2k+1\geq [n/2].$ Therefore, $T$ has one element of $\zn$ repeating at least $[n/2]$ times. Then by Lemma 3.1, we get the required zero subsequence of length $n.$ $\hfill\Box$ \bigskip \noindent{\bf N-C Extension 2.} {\it Let $n$ and $k$ be positive integers such that $1 \leq k < (n+2)/3.$ Let $S$ be a minimal zero sequence of $\zn$ of length $n-k+1.$ Then there exists $a\in\zn$ such that $a$ appears in $S$ at least $n-2k+2$ times.} \smallskip \noindent{\bf Proof.} This is nothing but (I) of Theorem 3.2. Since in Theorem 3.2, it has been proved that (I) $\iff$ (II) $\iff$ (III) and by Theorem 3.3, we get the result. $\hfill\Box$ \bigskip For the simillar reasons, the following two statements are also true. \bigskip \noindent{\bf N-C Extension 3.} {\it Let $n$ and $k$ be positive integers such that $n-2k\geq 1.$ Let $S$ be a zero-free sequence in $\zn$ of length $n-k.$ Then one element of $S$ is repeated at least $n-2k+1$ times.} \bigskip \noindent{\bf N-C Extension 4.} {\it Let $n$ and $k$ be positive integers such that $n-2k\geq 1.$ Let $S$ be a sequence of $\zn$ of length $2n-k-1.$ Suppose $S$ does not have a zero subsequence of length $n.$ Then there exist $a \ne b \in \zn$ such that $a$ and $b$ appear in $S$ at least $n-2k+1$ times.} \bigskip The statement (III) in Theorem 3.2 is the generalization of the following results. \bigskip \noindent{\bf Corollary 3.2.1} {\it Any sequence $S$ in $\zn$ of length $2n-2$ having no zero subsequence of length $n$ consists of two distinct elements in $\zn$ each appearing exactly $n-1$ times.} \smallskip \noindent{\bf Proof.} Put $k = 1$ in the statement (III) in Theorem 3.2, we get the result. $\hfill\Box$ \bigskip Corollary 3.2.1 was first proved by Yuster and Peterson (cf. \cite{yp1}) and also proved by Bialostocki and Dierker (cf. \cite{bd1}). When $k = 2$ in Theorem 3.2, we get any sequence in $\zn$ of length $2n-3$ which does not have any zero subsequence of length $n$ will have two distinct elements of $\zn$ each appearing at least $n-3$ times. Indeed, a better result was proved by C. Flores and O. Ordaz (cf. \cite{fo1}) as follows. \bigskip \noindent{\bf A Result of Flores and Ordaz.} (cf. \cite{fo1}) {\it Suppose $S$ is any sequence in $\zn$ of length $2n-3$ such that $S$ has no zero subsequence of length $n$. Then there exists $a, b \in \zn$ such that $\zn$ is generated by $b-a$ and $a$ appearing $n-1$ times in $S$ and one of the following conditions hold; (i) $b$ appearing exactly $n-2$ times. (ii) $b$ appearing exactly $n-3$ times in $S$ and also, $2b -a$ appearing exactly once in $S.$} \bigskip \noindent{\it Remark 2} (i) By putting $k =1$ in the statement (II) of Theorem 3.2, we get, if $S$ is a zero-free sequence in $\zn$ of length $n-1,$ then $S$ consists of only one element $a \in \zn$ which is appearing $n-1$ times. Since $S$ is zero-free, it is clear that the order of $a$ has to be $n.$ (ii) Now, we put $k = 2$ in the statement (II) in Theorem 3.2. If a zero-free sequence $S$ in $\zn$ of length $n-2,$ we get less information about the structure of $S.$ Using the following result of Hamidoune, we see in the following Proposition 3.4, in this case, the structure of $S.$ \bigskip \noindent{\bf A Result of Hamidoune.} (See for instance, Lemma 2.3 in \cite{fo1}) {\it Let $S$ be a zero-free sequence in $\zn$ of length at least $n-2.$ Also assume that $S$ consists of at most $2$ distinct residue modulo $n.$ Then the length of $S$ is at most $n-1$ and will have one of the following form; (i) $S : {\underbrace{a, a, \cdots, a}_{r \ {\rm times}}}$ where $r \leq n-1.$ (ii) $S : {\underbrace{a, a, \cdots, a}_{n-2 \ {\rm times}}}, 2a.$} \bigskip Using this, we shall prove the following Proposition. \bigskip \noindent{\bf Proposition 3.4} {\it Let $S$ be a zero-free sequence in $\zn$ of length $n-2.$ Then $S$ consists of an element $a \in \zn$ which is repeated either $n-2$ times or $n-3$ times and $2a$ appearing exactly once.} \smallskip \noindent{\bf Proof.} By putting $k = 2$ in the statement Theorem 3.2, we get the sequence $S$ has one element $a \in \zn$ which is repeated at least $n-3$ times. If $a$ is repeated $n-2$ times, then nothing to prove. If $a$ is repeated $n-3$ times exactly, then $S$ can consist only of two distinct residue classes modulo $n.$ Then by above mentioned result of Hamidoune, it follows that the second residue has to be $2a$ and the result follows. $\hfill\Box$ \bigskip At this juncture, we should mention a beautiful result of Gao and Geroldinger (cf. \cite{gg11}) as follows. \bigskip \noindent{\bf A result of Gao and Geroldinger.} {\it Let $n \geq 4$ be a natural number. Let $S$ be a zero-free sequence in $\zn$ of length at least $(n+3)/2.$ Then, there exists $a\in \zn$ of order $n$ and $a$ appears in $S$ at least $n/6 + 13/12$ times.} \bigskip Using this result and the above techniques, we can prove the following theorem (we skip the proof here). \bigskip \noindent{\bf Theorem 3.5} {\it Let $S$ be a sequence in $\zn$ of length at least $n-1+(n+3)/2.$ Suppose $S$ does not have any zero subsequence of length $n.$ Then there exist $a \ne b \in \zn$ such that both appears in $S$ at least $n/6 + 13/12$ times and either $a$ or $b$ is of order $n.$ } \bigskip \noindent{\bf Theorem 3.6} {\it Let $n,k$ be positive integers such that $n-2k \geq 1.$ Let $S$ be a sequence in $\zn$ of length $2n-k-1$ such that at most one element in $S$ appears more than $n-2k$ times. Then there exists a zero subsequence of $S$ of length $n.$} \smallskip \noindent{\bf Proof.} Suppose not, that is $S$ does not have zero subsequence of length $n.$ Then by the statement (III) in Theorem 3.2 we get a contradiction and hence the result. $\hfill\Box$ \bigskip We give a short proof of the following known theorem. \bigskip \noindent{\bf Theorem 3.7} {\it Let $S$ be a sequence in $\zn$ of length $2n-1.$ The sequence $S$ has exactly one zero subsequence of length $n.$ in $\zn$ if and only if there exists $a$ and $b$ in $\zn$ such that $a$ appears $n$ times and $b$ appears $n-1$ times in $S.$} \smallskip \noindent{\bf Proof.} The converse is easy. We shall prove the other implication. Let $$S : a_1, a_2, \cdots, a_{2n-1}$$ be the given sequence. By EGZ theorem, there is a zero subsequence of length $n.$ We let $a_1+a_2+\cdots+a_n \equiv 0\pmod{n},$ if necessary by renaming the indices. Let $S_1$ be a subsequence of $S$ of length $2n-2$ such that $a_1$ does not appear in $S_1.$ Clearly by hypothesis, $S_1$ does not have any zero subsequence of length $n.$ Therefore by Corollary 3.2.1, $S_1$ consists of two distinct elements $a$ and $b$ of $\zn$ each appearing $n-1$ times. Now, let us consider a subsequence $S_2$ of $S$ such that $a_n=a$ does not appear in $S_2.$ Clearly, $S_2$ has length $2n-2$ and it cannot have a zero subsequence of length $n,$ by the hypothesis. Therefore, it forces that $a_1$ has to be $a$ and hence the theorem. $\hfill\Box$ \section*{\normalsize 4. Concluding Remarks} \addtocounter{section}{+1} In this section, we shall discuss about the analoguous situation of the non-canonical extensions of EGZ Theorem for the group $\zp\oplus\zp.$ For the detail history and status of the following conjectures, one may refer to \cite{th1}. \bigskip There is no result so far known which is an analogue of N-C Extension 1 for the group $\zp\oplus\zp.$ The analogue of EGZ Theorem for the group $\zp\oplus\zp$ is the following conjecture of Kemnitz. \bigskip \noindent{\bf Conjecture 4.1} (Kemnitz, 1983, \cite{ke1}) {\it Let $S$ be a sequence in $\zp\oplus \zp$ of length $4p-3.$ Then there exists a zero subsequence $T$ of $S$ of length $p.$} \bigskip Conjecture 4.1 is yet to be resolved. For the analogue of Corollary 3.2.1, we have the following conjecture of Gao. \bigskip \noindent{\bf Conjecture 4.2} (W. D. Gao, 2000, \cite{gao}) {\it Let $S$ be sequence in $\zp\oplus\zp$ of length $4p-4.$ Suppose that $S$ does not contain any zero subsequences of length $p.$ Then $S$ consists of four distinct elements $a, b, c$ and $d$ in $\zp\oplus\zp$ each of them appearing in $S$ exactly $p-1$ times.} \bigskip Conjecture 4.2 is also open. The analogue of Remark 2 (i) is the following conjecture of van Emde Boas. \bigskip \noindent{\bf Conjecture 4.3} (van Emde Boas, 1969, \cite{veb1}) {\it Let $S$ be a sequence in $\zp\oplus\zp$ of length $3p-3.$ If $S$ does not contain any zero subsequences of length at most $p,$ then $S$ consists of three distinct elements $a, b$ and $c$ of $\zp \oplus\zp$ each appearing exactly $p-1$ times.} \bigskip Of course, Conjecture 4.3 is also open till now. The analogue of the statement (I) in Theorem 3.2 when $k =1$ is the following conjecture of Gao and Geroldinger. \bigskip \noindent{\bf Conjecture 4.4} (Gao and Geroldinger, 1998, \cite{gg12}) {\it If $S$ is a minimal zero sequence in $\zp\oplus\zp$ of length $2p-1,$ then there exists $a \in \zp\oplus\zp$ such that $a$ is appearing in $S$ at least $p-1$ times.} \bigskip As we have seen in the last section, it natural to ask for the inter relationships between these four conjectures. \bigskip Conjecture 4.2 implies Conjecture 4.1 was proved by Gao (cf. \cite{gao}). Conjecture 4.4 implies Conjecture 4.3 was proved by Gao and Geroldinger (cf. \cite{gg12}). Conjecture 4.2 implies Conjecture 4.3 was proved by the author (cf. \cite{th1}). Also, in \cite{th1} it has been proved the following partial implication. \bigskip \noindent{\bf Theorem 4.4} (Thangadurai, 2001, \cite{th1}) {\it Assume Conjecture 4.3. Let $S$ be a sequence of the form $${\underbrace{a, a, \cdots, a,}_{s \ {\rm times}}} a_1, a_2, \cdots, a_{4p-4-s}$$ where $a, a_i \in \zp\oplus\zp$ for all $i = 1, 2, \cdots, 4p-4-s$ and $s > \left[\frac{p-3}{2}\right].$ Suppose $S$ does not contain a zero subsequence of length $p.$ Then $S$ consists of four distinct elements $a, b, c$ and $d$ in $\zp\oplus\zp$ each of them appearing exactly $p-1$ times. In other words, $S$ satisfies Conjecture 4.2.} \begin{thebibliography}{99} \bibitem{al1} N. Alon, { Subset sums}, {\it J. Number Theory}, {\bf 27} (1987), 196-205. \bibitem{al2} N. Alon, A. Bialostocki and Y. Caro, The extremal cases in Erd\H{o}s-Ginzburg-Ziv theorem, Unpublished. \bibitem{an} D. D. Anderson, {\it Factorization in integral domains}, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, {\bf 189}, 1997. \bibitem{bl1} A. Bialostocki and M. Lotspeich, {Some developments on the Erd\H{o}s-Ginzburg-Ziv theorem I.}, {\it Sets, graphs and numbers (Butapest, 1991)}, 97-117, Colloq. Math. Soc. J\'anos Bolyai, 60, North-Holland, Amsterdam, 1992. \bibitem{bd1} A. Bialostocki and P. Dierker, {On Erd\H{o}s-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings}, {\it Discrete Mathematics}, {\bf 110} (1992), 1-8. \bibitem{ben1} J. D. Bovey, P. Erd\H{o}s and I. Niven, {Conditions for zero-sum modulo $n,$} {\it Canad. Math. Bull.}, {\bf 18} (1975) 27-29. \ \bibitem{ca1} Y. Caro, { Zero-sum problems -- A survey}, {\it Discrete Mathematics} {\bf 152}(1996), 93-113. \bibitem{ca2} Y. Caro, On zero-sum Ramsey Numbers--Stars, {\it Discrete Mathematics} {\bf 104} (1992) 1-6. \bibitem{cu1} A. L. Cauchy, Recherches sur les nombers, {\it J. Ecole Polytechniques}, {\bf 9} (1813), 99-123. \bibitem{da1} H. Davenport, On the addition of residue classes, {\it J. London Math. Soc.}, {\bf 22} (1947), 100-101. \bibitem{dm} G. T. Diderrich and H. B. Mann, Combinatorial problems in finite abelian groups, in: {\it A survey of combinatorial theory}, edited by J. N. Srivastava, North-Holland, 1973. \bibitem{egz} P. Erd\"os, A. Ginzburg and A. Ziv, { Theorem in the additive number theory}, {\it Bull. Res. Council Israel}, {\bf 10 F}(1961), 41-43. \bibitem{eg} P. Erd\"os and R. L. Graham, {\it Old and new problems and results in combinatorial number theory}, Monographie N. 28 de L'Enseignement Math\'ematique, Univerit\'e de Gen\'eve, 1980. \ \bibitem{fo1} C. Flores and O. Ordaz, On sequences with zero sum in Abelian groups. Volume in homage to Dr. Rodolfo A. Ricabarra (Spanish), 99--106, Vol. Homenaje, 1, Univ. Nac. del Sur, Bahía Blanca, 1995. \ \bibitem{ga8} W. D. Gao, { Addition theorems for finite abelian group}, {\it J. Number Theory}, {\bf 53}(1995), 241-246. \bibitem{ga1} W. D. Gao, { On zero-sum subsequences of restricted size}, {\it J. Number Theory}, {\bf 61}(1996), 97-102. \bibitem{ga5} W. D. Gao, {A combinatorial problem on finite Abelian group}, {\it J. Number Theory} {\bf 58} (1996), 100-103. \bibitem{ga6} W. D. Gao, {An addition theorem for finite cyclic groups}, {\it Discrete Math.}, {\bf 163} (1997), no. 1-3, 257-265. \bibitem{ga7} W. D. Gao, {On the number of zero sum subsequences}, {\it Discrete Math.} 163 (1997), no. 1-3, 267--273. \bibitem{ga2} W. D. Gao, { Addition Theorems and Group Rings}, {\it J. Comb. Theory}, Series A, (1997), 98-109. \bibitem{gao} W. D. Gao, { Two zero-sum problems and multiple properties}, {\it J. Number Theory}, {\bf 81}(2000), 254-265. \bibitem{gg11} W. D. Gao and A. Geroldinger, {On the structure of zero-free sequences}, {\it Combinatorica}, {\bf 18} (1998), 519-527. \bibitem{gg12} W. D. Gao and A. Geroldinger, {On long minimal zero sequences in finite abelian groups}, {\it Periodica Mathematica Hungarica}, Vol. 38 (3)(1999), 179-211. \bibitem{gy1} W. D. Gao and Y. X. Yang, {Note on a combinatorial constant}, {\it J. Math. Res. and Expo.} {\bf 17} (1997), 139-140. \bibitem{ke1} A. Kemnitz, { On a lattice point problem}, {\it Ars Combinatorica}, {\bf 16b}(1983), 151-160. \bibitem{ma1} H. B. Mann, {Two addition theorems}, {\it J. Combinatorial Theory}, {\bf 3} (1967), 233-235. \bibitem{man1} H. B. Mann, {\it Addition theorems: the addition theorems of group theory and number theory}, Interscience Publishers, J. Wiley $\&$ Sons, 1965. \bibitem{man2} H. B. Mann, {Additive group theory- a progress report,} {\it Bull. Amer. Math. Soc.,} {\bf 79} (1973), 1069-1075. \bibitem{na} M. B. Nathanson, {\it Additive Number Theory}, GTM {\bf 165}, Springer, 1996. \bibitem{o1} J. E. Olson, { On a combinatorial problem of finite Abelian groups I and II}, {\it J. Number Theory}, {\bf 1}(1969), 8-10, 195-199. \bibitem{o2} J. E. Olson, { On a combinatorial problem of Erd\"os, Ginzburg and Ziv}, {\it J. Number Theory}, {\bf 8}(1976), 52-57. \bibitem{yp1} B. Peterson and T. Yuster, {A generalization of an addition theorem for solvable groups}, {\it Can. J. Math.}, vol XXXVI, No. 3 (1984) 529-536. \bibitem{z1} Shan Zun, {A conjecture in elementary number theory}, {\it Adv. Math. (Beijing)}, {\bf 12}, No. 4, (1983), 299-301. \bibitem{su1} B. Sury, { The Chevalley-Warning theorem and a combinatorial question on finite groups}, {\it Proc. Amer. Math. Soc.}, {\bf 127} (1999), 4, 951-953. \bibitem{th1} R. Thangaduari, Interplay between four conjectures on certain zero-sum problems, To appear in: {\it Expositiones Math.} \bibitem{veb1} P. van Emde Boas, A combinatorial problem on finite Abelian groups II, Z. W. (1969-007) Math. Centrum-Amsterdam. \end{thebibliography} \end{document}