Part II, beginning with Chapter 7, initiates the material on zero-sum applications. We remarked in Chapter 2 that the Erd˝os-Ginzburg-Ziv Theorem can be viewed as a gener- alization of the pigeonhole principle. Having discovered a zero-sum generalization for the pigeonhole principle, one of the simplest Ramsey-type extremal problems, it was natural to wonder if a similar generalization might also occur for more complex extremal questions.
One particular incarnation of this idea can be described as follows. Let Lt
m be a fixed
system of inequalities (often linear) in tm variables x1
1, . . . , x1m, x21, . . . , x2m, . . . , xt1, . . . , xtm,
let f(Ltm, r) denote the minimal integer N such that no matter how the first N integers, denoted [1, N], are r-colored, say by ∆ : [1, N] → {0,1, . . . , r−1}, there will always be an integer solution to Lt
m, given by xji = tji, with ∆(t1j) = ∆(tj2) = . . . = ∆(tjm), for
each j = 1, . . . , t (we call such a solution monochromatic). Likewise, let fzs(Ltm,2) denote
the minimal integer N such that no matter how the first N integers are colored using the elements from Z/mZ, say by ∆ : [1, N] → Z/mZ, there will always be an integer solution to Ltm, given by xji = tji, such that Pm
i=1
∆(tji) = 0, for each j = 1, . . . , t (we call such a solution zero-sum). In short, the previous extremal functions involve looking for a collection of m-uniform (in number of elements) solutions to a (usually linear) system of inequalities, each individually monochromatic or zero-sum, respectively, with additional inequality relations required amongst the t solutions of size m. Note that EGZ says that
f(P1
m,2) =fzs(Pm1,2) = 2m−1, wherePm1 is the systemx1< x2 < . . . < xm inm variables
(i.e., no restriction on the variables except that they be distinct in value).
Since we are allowed to use only 0’s and 1’s in the coloring for fzs(Ltm,2) (in which
monochromatic solutions), it is easily seen that a lower bound construction for f(Lt m,2)
yields a lower bound construction for fzs(Ltm,2). Hence f(Ltm,2) ≤ fzs(Ltm,2). On the
other hand, since a monochromatic solution is always zero-sum, we have the inequality
fzs(Ltm,2)≤f(Ltm, m). If the first inequality is an equality, i.e., f(Ltm,2) = fzs(Ltm,2), as
it is for instance for EGZ, then we say the system Lt
m zero-sum generalizes. In essence,
the system zero-sum generalizing means that best way to avoid zero-sum solutions is to avoid monochromatic solutions. One might at first think this a very unusual occurrence, particularly since there is such additional freedom when coloring with Z/mZversus{0,1}; however many examples attaining equality have been found. Though no formal proof or theorem is known, it is generally believed (by at least some) that as long as the restrictions on the variables are ‘sufficiently nonrestrictive,’ such a zero-sum generalization will occur.
One might wonder if there is a natural way to obtain a zero-sum generalization for
f(Lt
m, r) when r >2. The easiest and most straightforward way is to simply replace pairs
of colors from {0,1, . . . , r−1} with disjoint copies ofZ/mZ, leaving intact an odd-person- out color (if r is odd). Formally, letfzs(Lt
m, r) denote the minimal integer N such that no
matter how the first N integers are colored using the elements from br2c disjoint copies of
Z/mZ and (ifr is odd) an additional disjoint color class, say by
∆ : [1, N]→(Z/mZ)(1)G(Z/mZ)(2)G. . .G(Z/mZ)(br2c),
ifr is even, or by
∆ : [1, N]→(Z/mZ)(1)G(Z/mZ)(2)G. . .G(Z/mZ)(br2c) t {∞}
ifris odd, then there will always be an integer solution to Lt
for each j = 1, . . . , t, we have both that ∆(e tj1) = ∆(e tj2) = . . . = ∆(e tjm), and that either
∆(tji) = ∞ for all i or else Pm
i=1
∆(tji) = 0, where ∆ : [1e , N] → {1, . . . ,dr2e} is the coloring given by∆(e t) =sif ∆(t)∈(Z/mZ)(s), and otherwise∆(e t) =dr2e. Then we once more have the inequalities f(Ltm, r)≤fzs(Lmt , r)≤f(Ltm,(m−1) jr 2 k + lr 2 m ),
and there is an r-color zero-sum generalization wheneverf(Lt
m, r) =fzs(Ltm, r).
Very few examples of r-color zero-sum generalizations with r > 2 are known—due (perhaps?) to the added difficulty of such problems—but there have been a handful of examples. One might lament that this definition for an r-color zero-sum generalization is somewhat unnatural, particularly in the odd case, and is thus not entirely satisfactory. On the positive side, this ‘weak’ notion of zero-sum generalization is defined for every r ≥ 2 and requires no machinery to show it is well defined. There is (sometimes) an alternative ‘strong’ notion of zero-sum generalization, that, though not dealt with in this thesis, is worth mentioning.
Let τ(m, s) be the maximal integer τ such that there exists a cardinality τ subset X
of Z/mZ×. . .×Z/mZ
| {z }
s
with the property that every m-term zero-sum subsequence with its terms fromX must be monochromatic. Letκ(m, s) be the minimal integerκ such that every sequence of terms fromZ/mZ×. . .×Z/mZ
| {z }
s
with lengthκ contains an m-term zero- sum subsequence. Observe that takingm−1 terms equal to each of the elements from the set X from τ(m, s) gives a lower bound for κ(m, s). Hence τ(m, s)(m−1) + 1 ≤κ(m, s). That equality holds fors= 1 follows from EGZ, while it is a recent result of C. Reheir [51], affirming the long-standing Kemnitz Conjecture, that equality holds fors= 2 as well. The
determination of κ(m, s) for s > 3 seems extremely difficult, and even the determination of τ(m, s) is quite nontrivial. However, whenever τ(m, s)(m−1) + 1 = κ(m, s), one can define fsz(Ltm, τ(m, s)) to be the minimal integer N such that no matter how the first N
integers are colored using the elements from Z/mZ×. . .×Z/mZ
| {z } s , say by ∆ : [1, N] → Z/mZ×. . .×Z/mZ | {z } s
, then there will always be an integer solution toLt
m, given byxji =tji,
such that Pm
i=1
∆(tji) = 0, for eachj= 1, . . . , t. Then we have a strongτ(m, s)-color zero-sum generalization wheneverfsz(Ltm, τ(m, s)) =f(Ltm, τ(m, s)).
When defined, the ‘strong’ notion of zero-sum generalization is a perhaps more pleasing notion of an r-color zero-sum generalization. On the negative side, it only gives an r-color zero-sum generalization forrequal to someτ(m, s), which increases geometrically ins; there are no current methods that seem anywhere near sufficient to handle the associated added difficulties (the determination of κ(m, s) just for s= 2 was considered a major triumph in itself); and it is still unclear, though no counter examples are yet known, to what extent
τ(m, s)(m−1)+1 =κ(m, s). Regardless, this thesis deals only with ‘weak’r-color zero-sum generalizations, and most examples will be in the caser = 2. Additionally, since the second part deals only with un-weighted zero-sum questions, the sequenceW in Theorems 3.1 and 3.2 is always assumed to be all 1’s.