11.1
Discussion
The original nondecreasing diameter problem of Bialostocki, Erd˝os and Lefmann, as well as the zero-sum problems from Chapters 7 and 10, all dealt with finding not just one zero-sum subset, but a pair of such sets, each individually zero-sum. However, in all cases the paired zero-sum subsets were disjoint. One might also wonder about zero-sum generalizations for multiplem-sets with a prescribed intersection structure.
If we think of the sequence S of length n (in which we are trying to find the collection of zero-sum subsequences) as being a Z/mZ-coloring of the vertices of the complete m- uniform hypergraph Km
n, then the edges ofKmn correspond to them-term subsequences of
S. A collection ofm-term subsequences with a prescribed intersection structure is then just somem-uniform hypergraphH, whose vertex set we denote byV(H), and whose edge set we denote by E(H). If every e∈ E(H) satisfies P
v∈e∆(v) = 0, then we say the m-uniform
hypergraph His zero-sum.
Armed with this notation, we can define what it would mean for a given m-uniform hypergraphHto zero-sum generalize. Letf(H) (letfzs(H)) be the least integernsuch that
for every 2-coloring (coloring with the elements ofZ/mZ) of the vertices ofKm
n, there exists
a subhypergraph K isomorphic to H such that every edge ein K is monochromatic (such thatK is zero-sum). From the pigeonhole principle it is clear thatf(H)≤2|V(H)|−1, with equality holding ifH is connected. Then them-uniform hypergraphHzero-sum generalizes if fzs(H) = f(H), which in the connected case simply means fzs(H) = 2|V(H)| −1. The
Erd˝os-Ginzburg-Ziv Theorem is then the statement that there is a zero-sum generalization for them-uniform hypergraph consisting of a single edge.
Not every hypergraph zero-sum generalizes. For instance, a completem-uniform hyper- graph onk > mvertices is easily seen to requirem(k−1)+1 vertices to guarantee a zero-sum copy of itself (which will necessarily be monochromatic). Note thatm(k−1) + 1>2k−1, form >2, and so no zero-sum generalization is present. However, the goal of this chapter is to show that a zero-sum generalization does occur provided the hypergraph has very little intersection structure. More concretely, we will be able to show a zero-sum generalization for anym-uniform hypergraph on two edges, and any hypergraph with ‘many’ monovalent vertices (vertices contained in precisely one edge). The proofs are simple applications of the combined machinery of Chapters 3, 4 and 6, and were the original motivation for developing the results from Chapters 4, 5 and 6.
11.2
EGZ in Hypergraphs
Theorem 11.1 below can be used to show a zero-sum generalization for an m-uniform hypergraph that can be iteratively constructed by first starting with a zero-sum generalizing hypergraph (like a single edge or pair of edges), and then adding edges, one by one, so that each added edge has—at the time of its addition—at least half its vertices monovalent.
Theorem 11.1. Let H be a finite m-uniform hypergraph, let e∈E(H), and let H0 be the subhypergraph obtained by removing the edge e and all monovalent vertices contained in
e. If fzs(H0) ≤ 2|V(H0)| −1 and e has at least dm2e monovalent vertices, then fzs(H) ≤
2|V(H)| −1.
Proof. Let S denote the sequence given by a coloring ∆ : V → Z/mZ, wheren =|V(H)| and V = V(Km
2n−1). Let s be the number of non-monovalent vertices in e. Note that by assumption s ≤ bm2c. We may assume that the multiplicity of each term in S is at most
n−1, else there will be a zero-sum copy ofHwith all edges monochromatic. Hence, if there exists a subset X ⊆V such that |X| ≤s−2 ≤ bm2c −2 and |∆(V \X)| ≤ 2, then setting asiden−m terms colored byai for each of the two ai ∈∆(V \X) and applying Theorem
4.1 to the remaining 2m−1 terms, it follows that there exists an edge-wise zero-sum copy of Hwith the vertices ofecolored by the zero-sum sequence given by Theorem 4.1 and all other edges monochromatic. Otherwise, sinces≤ bm
2c, then it follows from Proposition 2.3 that there exists an (2n−m)-set partition P0 of S with at least 2n−2m+s cardinality
one sets. Let P be the (m−s)-set partition obtained from P0 by removing 2n−2m+s
cardinality one sets. Since s≤ bm
2c, it follows that m−s≥ m2 −1, whence we can apply Theorem 3.2 to P, yielding two cases.
If Theorem 3.2(i) holds, then let Abe the corresponding (m−2)-set partition given by (i). Applying Theorem 6.1 to the set partition A yields an (m−s)-set partition A0 that
contains at most 2(m−s) terms of S, and whose sumset is Z/mZ. This leaves at least 2n−1−2(m−s) = 2(n−m+s)−1≥2|V(H0)| −1 vertices not contained in any term of
A0. Thus, since f
zs(H0) ≤2|V(H0)| −1, it follows that there exists an edge-wise zero-sum
copy of H0 not containing any vertices contained in A0. Hence, since the sumset of terms
which together with the vertices of H0 form an edge-wise zero-sum copy of H.
If Theorem 3.2(ii) holds, then there exists a proper nontrivial subgroup Ha of indexa
such that all but at most a−2 terms of S are from the coset α +Ha, and w.l.o.g. by
translation we may assumeα= 0; furthermore, there exists a subsequenceS0 ofSof length
at mostm−s+ma−1 with an (m−s)-set partitionP0 =P0
1, . . . , Pm0 −ssatisfying mP−s
i=1
P0
i =Ha.
Hence it follows that there are at least 2n−1−(m−s+m
a −1)−(a−2)≥2n−1−2(m−s)
terms of S that are not used in the set partition P0, and which are from H
a, whence the
proof is complete as it was in the previous paragraph.
A simple corollary of Theorem 11.1 is the following result.
Theorem 11.2. LetHbe a connected, finitem-uniform hypergraph. If every subhypergraph
H0 ofHcontains an edge with at least half of its vertices monovalent inH0, thenHzero-sum generalizes.
Proof. IfHhas one edge, this is precisely a restatement of the Erd˝os-Ginzburg-Ziv Theorem. Hence the upper bound for Theorem 11.2 follows from Theorem 11.1 and induction on the number of edges (relaxing the connectedness condition), while the lower bound for connected His trivial.
The final zero-sum generalizing result of this section will require the following simple proposition, easily proved by induction ons.
Proposition 11.3. Let m and s be positive integers, and let S be a sequence of elements
from an abelian group of order m. If |S| ≥m+ 2s−1, then there exist two disjoint s-term subsequences ofS whose sums are equal.
Theorem 11.4. If H is a hypergraph that consists of two intersecting m-sets, then H
Proof. Let S denote the sequence given by a coloring ∆ :V → Zm, where n=|V(H)|and
V =V(Km
2n−1). Let the two edges of H be A and B. If |A∩B|<dm2e, then the proof is complete by Theorem 11.2. So we may assume |A∩B| ≥ dm2e. Let s=m− |A∩B|. Note
n=m+s,|S|= 2m+ 2s−1, ands≤ bm
2c.
We may also assume that the multiplicity of each term inS is at mostn−1, else there will be a zero-sum copy ofH with all edges monochromatic. Hence, if there exists a subset
X⊆V such that|X| ≤ dm2e−2 and|∆(V\X)| ≤2, then setting asidesterms colored byai
for each of the twoai∈∆(V\X), and applying Theorem 4.1 to the remaining 2m−1 terms,
it follows that there exists an edge-wise zero-sum copy of Hwith the vertices of A colored by the zero-sum sequence given by Theorem 4.1, and withV(H)\(A∩B) monochromatic. Otherwise, it follows from Proposition 2.3 that there exists an (m+ 2s)-set partitionP0 of
S with at leastdm
2e+ 2scardinality one sets. LetP be thebm2c-set partition obtained from
P0 by removingdm
2e+ 2scardinality one sets. Applying Theorem 3.2 toP yields two cases. If Theorem 3.2(i) holds, then letA0 be the set partition given by (i). Applying Theorem 6.1 to the set partitionA0 yields anbm
2c-set partitionA00 that contains at mostmterms of
S, and whose sumset is Z/mZ. This leaves at least m+ 2s−1 vertices not contained in any term ofA00. Hence from Proposition 11.3, it follows that there are two disjoints-term subsequences S1 and S2, none of whose terms are contained in a term of A00, and whose sums are equal to (say)t. Sinces≤ bm2c, then letT be a subsequence of lengthm−s− bm2c whose terms are not contained inS1,S2, nor any term ofA00. Lett0 be the sum of the terms inT ifT is nonempty, and otherwise let t0 = 0. Sinces≤ bm
2c, and since the sumset of A00 is Z/mZ, it follows that we may choose bm2c terms of S from A00 whose sum is −(t+t0),
which along withS1,S2 and T yields a zero-sum copy ofHwith the terms fromA00and T contained inA∩B.
If Theorem 3.2(ii) holds, then there exists a proper nontrivial subgroup Ha of index
a such that all but at most a−2 terms of S are from the coset α +Ha, and w.l.o.g.
by translation we may assume α = 0; furthermore, there exists a subsequence S0 of S
of length at most bm
2c+ ma −1 with an bm2c-set partition P0 = P10, . . . , Pb0m2c satisfying bPm2c
i=1
P0
i = Ha. Hence, since dm2e ≤ m−s, then by appending on m−s− bm2c singleton
sets to P0, each with their element fromH
a, it follows that there exists a subsequenceS00
of S0, satisfying |S00| ≤ m−s+ m
a −1, and which has an (m−s)-set partition P00 the
sumset of whose terms is Ha (that there are enough terms from Ha to accomplish this
follows from the calculation of the next sentence). Hence it follows that there are at least 2m+ 2s−1−(a−2)−(m−s+ ma −1) =m+ 3s− ma −a+ 2≥ ma + 2s−1 >0 terms of S that are not used in the set partitionP00 and which are fromH
a, whence the proof is
complete as it was in the previous paragraph.
We remark that the arguments used in this section to obtain upper bounds for colorings withZ/mZwork equally well for colorings with any abelian groupG of orderm, although in the noncyclic case the matching lower bound constructions do not hold.
We conclude by giving an example of a fairly simple hypergraph on (bm
2c+ 3)(dm2e −1) vertices with every edge having at least dm2e −2 monovalent vertices, but which does not zero-sum generalize, showing that thedm2e bound given in Theorems 11.1 and 11.2 can be improved at best to dm
2e −1. Let X be a set of bm2c+ 3 vertices, and for each bm2c+ 2 subsetX0 ofX, define an edge of the hypergraphHto beX0along withdm
2e−2 monovalent vertices disjoint from X. For the coloring of the complete graph, let ∆ consist entirely of an equal number of vertices colored by 0 and 1, and one vertex colored by dm
since the only non-monochromatic m-term zero-sum sequence is (0| {z }, . . . ,0 dm 2e−1 ,1| {z }, . . . ,1 bm 2c , lm 2 m ), (11.1)
it follows that any zero-sum copyH0 ofHmust have one of its edges, saye, use the coloring
given by (11.1). Since|e∩X|=bm2c+ 2, then it follows from the pigeonhole principle that
e∩Xmust contain an elementxcolored by 1 as well as an elementycolored by 0. However, from the definition ofH and ∆ we can then find an edge ofH0 that contains bothx andy
but not the single element colored bydm
2e, which, since there can be no non-monochromatic zero-sum edge using only the colors 0 and 1, cannot be zero-sum, contradicting that H0 is