Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdős more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [S03b], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdős-Ginzburg-Ziv theorem in the following way: If { as(mod ns)}s=1k covers each integer either exactly 2q − 1 times or exactly 2q times where q is a prime power, then for any c1,...,ck ∈ ℤ/qℤ there exists an I ⊆ {1,...,k} such that ∑s∈I 1/ns = q and ∑s∈Ics = 0. The main theorem of this paper unifies many results in the two realms and also implies an extension of the Alon-Friedland-Kalai result on regular subgraphs.

Zero-sum Ramsey theory is a newly established area in combinatorics. It brings to ramsey theory algebric tools and algebric flavour. The paradigm of zero-sum problems can be formulated as follows: Suppose the elements of a combinatorial structure are mapped into a finite group K. Does there exists a prescribed substructure the sum of the weights of its elements is 0 in K? We survey the algebric background necessary to develop the first steps in this area and its short history dated back to a 1960 theorem of Erdos-Ginzburg and Ziv. Then a sys­ tematic survey is made to encompass most of the results published in this area until 1.1.95. Several conjectures and open problems are cited along this manuscript with the hope to catch the eyes of the interested reader.

We study progression-free sets in the abelian groups \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=({{\mathbb {Z}}}_m^n,+)$$\end{document}G=(Zmn,+). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_k({{\mathbb {Z}}}_m^n)$$\end{document}rk(Zmn) denote the maximal size of a set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S \subset {{\mathbb {Z}}}_m^n$$\end{document}S⊂Zmn that does not contain a proper arithmetic progression of length k. We give lower bound constructions, which e.g. include that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_3({{\mathbb {Z}}}_m^n) \ge C_m \frac{((m+2)/2)^n}{\sqrt{n}}$$\end{document}r3(Zmn)≥Cm((m+2)/2)nn, when m is even. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=4$$\end{document}m=4 this is of order at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3^n/\sqrt{n}\gg \vert G \vert ^{0.7924}$$\end{document}3n/n≫|G|0.7924. Moreover, if the progression-free set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subset {{\mathbb {Z}}}_4^n$$\end{document}S⊂Z4n satisfies a technical condition, which dominates the problem at least in low dimension, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|S|\le 3^n$$\end{document}|S|≤3n holds. We present a number of new methods which cover lower bounds for several infinite families of parameters m, k, n, which includes for example: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_6({{\mathbb {Z}}}_{125}^n) \ge (85-o(1))^n$$\end{document}r6(Z125n)≥(85-o(1))n. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_3({{\mathbb {Z}}}_4^n)$$\end{document}r3(Z4n) we determine the exact values, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \le 5$$\end{document}n≤5, e.g. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_3({{\mathbb {Z}}}_4^5)=124$$\end{document}r3(Z45)=124, and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_4({{\mathbb {Z}}}_4^n)$$\end{document}r4(Z4n) we determine the exact values, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \le 4$$\end{document}n≤4, e.g. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_4({{\mathbb {Z}}}_4^4)=128$$\end{document}r4(Z44)=128. With regard to affine caps, i.e. sets without 3 points on a line, the new methods asymptotically improve the known lower bounds, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=4$$\end{document}m=4 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=5$$\end{document}m=5: in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {Z}}}_4^n$$\end{document}Z4n from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.519^n$$\end{document}2.519n to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3-o(1))^n$$\end{document}(3-o(1))n, and when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=5$$\end{document}m=5 from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.942^n$$\end{document}2.942n to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3-o(1))^n$$\end{document}(3-o(1))n. This last improvement modulo 5 appears to be the first asymptotic improvement of any cap in AG(n, m), when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 5$$\end{document}m≥5 over a tensor lifting from dimension 6 (see Edel, in Des Codes Crytogr 31:5–14, 2004).

We study edge-labelings of the complete bidirected graph ↔ Kn with functions from the set [d] = {1, . . . , d} to itself. We call a cycle in ↔ Kn a fixed-point cycle if composing the labels of its edges results in a map that has a fixed point, and we say that a labeling is fixed-point-free if no fixed-point cycle exists. For a given d, we ask for the largest value of n, denoted Rf (d), for which there exists a fixed-point-free labeling of ↔ Kn. Determining Rf (d) for all d > 0 is a natural Ramsey-type question, generalizing some well-studied zero-sum problems in extremal combinatorics. The problem was recently introduced by Chaudhury, Garg, Mehlhorn, Mehta, and Misra, who proved that d ≤ Rf (d) ≤ d +d and showed that the problem has close connections to EFX allocations, a central problem of fair allocation in social choice theory. In this paper we show the improved bound Rf (d) ≤ d, yielding an efficient (1− ε)-EFX allocation with n agents and O(n) unallocated goods for any constant ε ∈ (0, 1/2]; this improves the bound of O(n) of Chaudhury, Garg, Mehlhorn, Mehta, and Misra. Additionally, we prove the stronger upper bound 2d − 2, in the case where all edge-labels are permulations. A very special case of this problem, that of finding zero-sum cycles in digraphs whose edges are labeled with elements of Zd, was recently considered by Alon and Krivelevich and by Mészáros and Steiner. Our result improves the bounds obtained by these authors and extends them to labelings from an arbitrary (not necessarily commutative) group, while also simplifying the proof.

We prove the following extension of the Erdős-Ginzburg-Ziv Theorem. Let m be a positive integer. For every sequence {ai}i∈I of elements from the cyclic group Zm, where |I| = 4m − 5 (where |I| = 4m − 3), there exist two subsets A, B ⊆ I such that |A ∩ B| = 2 (such that |A ∩B| = 1), |A| = |B| = m, and ∑ i∈A ai = ∑ i∈b bi = 0.

We present a general algebraic technique and discuss some of its numerous applications in combinatorial number theory, in graph theory and in combinatorics. These applications include results in additive number theory and in the study of graph colouring problems. Many of these are known results, to which we present unified proofs, and some results are new.

We present a general algebraic technique and discuss some of its numerous applications in Combinatorial Number Theory, in Graph Theory and in Combinatorics. These applications include results in additive number theory and in the study of graph coloring problems. Many of these are known results, to which we present unified proofs, and some results are new.

Two conjectures concerning the Erdős Ginzburg Ziv theorem were recently confirmed. Reiher and di Fiore proved independently the two dimension analogue of the EGZ theorem, as conjectured by Kemnitz, and Grynkiewicz proved the weighed generalization of the EGZ theorem as conjectured by Caro. These developments trigger some further problems. First, we will present computer experiments that at least for small numbers reveal very simple phenomena of zero sum theorems that seem to be difficult to prove. Next, we will examine the notion of generalization of Ramsey type theorems in the sense of a given zero sum theorem in view of the new developments.

The modular subset sum problem consists of deciding, given a modulus $m$, a multiset $S$ of $n$ integers in $0..m$, and a target integer $t$, whether there exists a subset of $S$ with elements summing to $t \pmod{m}$, and to report such a set if it exists. We give a simple $O(m \log m)$-time with high probability (w.h.p.) algorithm for the modular subset sum problem. This builds on and improves on a previous $\tilde{O}(m)$ w.h.p. algorithm from Axiotis, Backurs, Jin, Tzamos, and Wu (SODA 19). Our method utilizes the ADT of the dynamic strings structure of Gawrychowski et. al (SODA 18). However, as this structure is rather complicated we present a much simpler alternative which we call the Data Dependent Tree. As an application, we consider the computational version of a fundamental theorem in zero-sum Ramsey theory. The Erdős-Ginzburg-Ziv Theorem states that a multiset of $2n - 1$ integers always contains a subset of cardinality exactly $n$ whose values sum to a multiple of $n$. We give an algorithm for finding such a subset in time $O(n \log n)$ w.h.p. which improves on an $O(n^2)$ algorithm due to Del Lungo, Marini, and Mori (Disc. Math. 09).

The cap set problem asks how large can a subset of ( Z / 3 Z ) n (\mathbb {Z}/3\mathbb {Z})^n be and contain no lines or, more generally, how can large a subset of ( Z / p Z ) n (\mathbb {Z}/p\mathbb {Z})^n be and contain no arithmetic progressions. This problem was motivated by deep questions about structure in the prime numbers, the geometry of lattice points, and the design of statistical experiments. In 2016, Croot, Lev, and Pach solved the analogous problem in ( Z / 4 Z ) n (\mathbb {Z}/4\mathbb {Z})^n , showing that the largest set without arithmetic progressions had size at most c n c^n for some c > 4 c > 4 . Their proof was as elegant as it was unexpected, being a departure from the tried and true methods of Fourier analysis that had dominated the field for half a century. Shortly thereafter, Ellenberg and Gijswijt leveraged their method to resolve the original cap set problem. This expository article covers the history and motivation for the cap set problem and some of the many applications of the technique: from removing triangles from graphs, to rigidity of matrices, and to algorithms for matrix multiplication. The latter application turns out to give back to the original problem, sharpening our understanding of the techniques involved and of what is needed for showing that the current bounds are tight. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, and the integers modulo N N .

The Kemperman structure theorem (KST) yields a recursive description of the structure of a pair of finite subsets A and B of an Abelian group satisfying |A + B| ≤ |A| + |B| - 1. In this paper, we introduce a notion of quasi-periodic decompositions and develop their basic properties in relation to KST. This yields a fuller understanding of KST, and gives a way to more effectively use KST in practice. As an illustration, we first use these methods to (a) give conditions on finite sets A and B of an Abelian group so that there exists b ∈ B such that |A + (B \ {b})| ≥ |A| + |B| - 1, and to (b) give conditions on finite sets A, B, C1,...,Cr of an Abelian group so that there exists b ∈ B such that |A + (B \ {b})| ≥ |A| + |B| - 1 and |A + (B \ {b}) + Σi=1r Ci| ≥ |A| |B| + Σi=1r |Ci| - (r + 2) + 1. Additionally, we simplify two results of Hamidoune, by (a) giving a new and simple proof of a characterization of those finite subsets B of an Abelian group G for which |A + B| ≥ min{|G| - 1, |A| + |B|} holds for every finite subset A ⊆ G with |A| ≥ 2, and (b) giving, for a finite subset B ⊆ G for which |A + B| ≥ min{|G|, |A| + |B| - 1} holds for every finite subset A ⊆ G, a nonrecursive description of the structure of those finite subsets A ⊆ G such that |A + B| = |A| + |B| - 1.

The Erdos-Ginzburg-Ziv Theorem [3] states that every sequence in an abelian group of order n having length 2n — 1 contains a subsequence with length n and zero sum. Several proofs of this result based on distinct ideas are contained in [1]. Olson generalized this result in [6], by showing that under the hypothesis n > 2 there is a non-null subgroup each element of which is a sum of a subsequence of length n, unless there is a value repeated (n + 1) times. Notice that Olson's Theorem applies also to non-abelian groups if one allows reordering of the sequence elements, (cf. [6]). A weighted generalization of the Erdos-Ginzburg-Ziv is conjectured in [2]. We consider in this paper some barycentric generalizations related to this conjecture. Our main result is the following. Let G be an abelian group of order n > 2. Let {x,: 1 < / ^ 2n — 1} be a family of elements of G with no (n + l)-repeated value and let {w, : 1 < i ^ n 1} be a family of integers relatively prime to n. Then there is a non-null subgroup H c G satisfying the following property. For every x € H there is a permutation x of [l,2n — 1] such that

The Erd\H{o}s-Ginzburg-Ziv constant $\mathfrak{s}(G)$ is the smallest $s$ such that every sequence of $s$ (not necessarily distinct) elements of $G$ has a zero-sum subsequence of length $\operatorname{exp}(G)$. For a prime $p$, let $r(\mathbb{F}_p^n)$ denote the size of the largest subset of $\mathbb{F}_p^n$ without a three-term arithmetic progression. Although similar methods have been used to study $\mathfrak{s}(G)$ and $r(\mathbb{F}_p^n)$, no direct connection between these quantities has previously been established. We give an upper bound for $\mathfrak{s}(G)$ in terms of $r(\mathbb{F}_p^n)$ for the prime divisors $p$ of $\operatorname{exp}(G)$. For the special case $G=\mathbb{F}_p^n$, we prove $\mathfrak{s}(\mathbb{F}_p^n)\leq 2p\cdot r(\mathbb{F}_p^n)$. Using the upper bounds for $r(\mathbb{F}_p^n)$ of Ellenberg and Gijswijt, this result improves the previously best known upper bounds for $\mathfrak{s}(\mathbb{F}_p^n)$ given by Naslund.

Summary.Let k ≥  1 be any integer. Let G be a finite abelian group of exponent n. Let sk(G) be the smallest positive integer t such that every sequence S in G of length at least t has a zero-sum subsequence of length kn. We study this constant for groups $$G \cong {\user2{{\mathbb{Z}}}}^{d}_{n}$$ when d = 3 or 4. In particular, we prove, as a main result, that $$s_{k} ({\user2{{\mathbb{Z}}}}^{3}_{p} ) = kp + 3p - 3$$ for every k ≥  4, $$5p + \frac{{p - 1}}{2} - 3 \leq s_{2} ({\user2{{\mathbb{Z}}}}^{3}_{p} ) \leq 6p - 3$$ and $$6p - 3 \leq s_{3} ({\user2{{\mathbb{Z}}}}^{3}_{p} ) \leq 8p - 7$$ for every prime p ≥  5.

Preliminaries Van der Waerden's theorem Supersets of $AP$ Subsets of $AP$ Other generalizations of $w(k r)$ Arithmetic progressions (mod $m$) Other variations on van der Waerden's theorem Schur's theorem Rado's theorem Other topics Notation Biobliography Index.

Let f(n, d) denote the least integer such that any choice of f(n, d) elements in $$ \mathbb{Z}^{d}_{n} $$ contains a subset of size n whose sum is zero. Harborth proved that (n-1)2d +1 ≤ f(n,d) ≤ (n-1)nd +1. The upper bound was improved by Alon and Dubiner to cdn. It is known that f(n-1) = 2n-1 and Reiher proved that f(n-2) = 4n-3. Only for n = 3 it was known that f(n,d) > (n-1)2d +1, so that it seemed possible that for a fixed dimension, but a sufficiently large prime p, the lower bound might determine the true value of f(p,d). In this note we show that this is not the case. In fact, for all odd n ≥ 3 and d ≥ 3 we show that $$ f{\left( {n,d} \right)} \geqslant 1.125^{{{\left[ {\frac{d} {3}} \right]}}} {\left( {n - 1} \right)}2^{d} + 1 $$.

Let S be a sequence of elements from the cyclic group Zm. We say S is zsf (zero-sum free) if there does not exist an m-term subsequence of S whose sum is zero. Denote by g(m, k) the least integer such that every sequence S with at least k distinct elements and length g(m, k) must contain an m-term subsequence whose sum is zero. Furthermore, denote by E(m, s) the set of all equivalence classes of zsf sequences with length s, up to order and affine transformation, that are not a proper subsequence of another zsf sequence. In this paper, we first find for a sequence S of sufficient length, |S| ≥ 2m − b 4 c − 2, necessary and sufficient conditions in terms of a system of inequalities over the integers for S to be zsf. Among the consequences, we determine g(m, k) for large m, namely g(m, k) = 2m− b k 2−2k+5 4 c provided m ≥ k − 2k − 3, which in turn resolves two conjectures of the first and fourth authors. Next, using independent methods, we evaluate g(m, 5) for every m ≥ 5. We conclude with the list of E(m, s) for every m and s satisfying 2m− 2 ≥ s ≥ max{2m− 8, 2m− b 4 c − 2}.

Let G be an additively written, finite abelian group, and exp(G ) its exponent. Let S=(a1 , ..., ak) be a sequence of elements in G; we say that S is a zero-sum sequence if i=1 ai=0. Let s(G ) be the samllest integer t such that every sequence of t elements in G contains a zero-sum subsequence of length exp(G ). This constant has been studied by serveral authors during last 20 years [1 9, 11]. Let Cn be the cyclic group of order n, and C n the direct product of k copies of Cn . In [3], Erdo s et al. proved that S(C n)=2n&1. A geometrical interpretation of s(C n) was given by Harborth in [8]. In 1980, Kemntiz suggested the following

Let G be a finite abelian group with exponent n. Let s(G) denote the smallest integer l such that every sequence over G of length at least l has a zero-sum subsequence of length n. For p-groups whose exponent is odd and sufficiently large (relative to Davenport’s constant of the group) we obtain an improved upper bound on s(G), which allows to determine s(G) precisely in special cases. Our results contain Kemnitz’ conjecture, which was recently proved, as a special case.

Let G be a finite abelian group and k ∈ N with k exp(G). Then Ek(G) denotes the smallest integer l ∈ N such that every sequence S ∈ F(G) with |S| ≥ l has a zero-sum subsequence T with k |T |. In this paper we prove that if G = Cn1 ⊕ · · · ⊕ Cnr is a p-group, k ∈ N with k exp(G) and gcd(p, k) = 1, then Ek(G) = ⌊ k k − 1 r ∑