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Graph Products, Fourier Analysis and Spectral Techniques
, 2003
"... We consider powers of regular graphs defined by the weak graph product and give a characterization of maximumsize independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In man ..."
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Cited by 24 (7 self)
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We consider powers of regular graphs defined by the weak graph product and give a characterization of maximumsize independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products. We show that the independent sets induced by the base graph are the only maximumsize independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size. Our approach is based on Fourier analysis on Abelian groups and on Spectral Techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on f0; : : : ; r \Gamma 1gn, generalizing some useful results from the f0; 1gn case.
A Lower Bound On The Mod 6 Degree Of The Or Function
 Computational Complexity
, 1995
"... We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in boolean variables over Zm . In particular, we say that a polynomial P weakly represents a boolean function f (both have n variables) if for any inputs x and y in f0; ..."
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Cited by 21 (1 self)
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We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in boolean variables over Zm . In particular, we say that a polynomial P weakly represents a boolean function f (both have n variables) if for any inputs x and y in f0; 1g n we have P (x) 6= P (y) whenever f(x) 6= f(y).
Zerosum sets of prescribed size
 Combinatorics, Paul Erdős is eighty
, 1993
"... Erdős, Ginzburg and Ziv proved that any sequence of 2n−1 integers contains a subsequence of cardinality n the sum of whose elements is divisible by n. We present several proofs of this result, illustrating various combinatorial and algebraic tools that have numerous other applications in Combinatori ..."
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Cited by 16 (4 self)
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Erdős, Ginzburg and Ziv proved that any sequence of 2n−1 integers contains a subsequence of cardinality n the sum of whose elements is divisible by n. We present several proofs of this result, illustrating various combinatorial and algebraic tools that have numerous other applications in Combinatorial Number Theory. Our main new results deal with an analogous multi dimensional question. We show that any sequence of 6n − 5 elements of Zn ⊕ Zn contains an nsubset the sum of whose elements is the zero vector and consider briefly the higher dimensional case as well. 1
Some Problems Involving RazborovSmolensky Polynomials
, 1991
"... Several recent results in circuit complexity theory have used a representation of Boolean functions by polynomials over finite fields. Our current inability to extend these results to superficially similar situations may be related to properties of these polynomials which do not extend to polyno ..."
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Cited by 11 (2 self)
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Several recent results in circuit complexity theory have used a representation of Boolean functions by polynomials over finite fields. Our current inability to extend these results to superficially similar situations may be related to properties of these polynomials which do not extend to polynomials over general finite rings or finite abelian groups. Here we pose a number of conjectures on the behavior of such polynomials over rings and groups, and present some partial results toward proving them. 1. Introduction 1.1. Polynomials and Circuit Complexity The representation of Boolean functions as polynomials over the finite field Z 2 = f0; 1g dates back to early work in switching theory [?]. A formal language L can be identified with the family of functions f i : Z i 2 ! Z 2 , where f i (x 1 ; : : : ; x i ) = 1 iff x 1 : : : x i 2 L. Each of these functions can be written as a polynomial in the variables x 1 ; : : : ; x n . We can consider algebraic formulas or circuits with...
On three zerosum Ramseytype problems
"... For a graph G whose number of edges is divisible by k, let R(G, Zk) denote the minimum integer r such that for every function f: E(Kr) ↦ → Zk there is a copy G ′ of G in Kr so that e∈E(G ′ ) f(e) = 0 (in Zk). We prove that for every integer k, R(Kn, Zk) ≤ n + O(k3 log k) provided n is sufficiently ..."
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Cited by 6 (4 self)
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For a graph G whose number of edges is divisible by k, let R(G, Zk) denote the minimum integer r such that for every function f: E(Kr) ↦ → Zk there is a copy G ′ of G in Kr so that e∈E(G ′ ) f(e) = 0 (in Zk). We prove that for every integer k, R(Kn, Zk) ≤ n + O(k3 log k) provided n is sufficiently large as a function of k and k divides � n 2 �. If, in addition, k is an odd primepower then R(Kn, Zk) ≤ n + 2k − 2 and this is tight if k is a prime that divides n. A related result is obtained for hypergraphs. It is further shown that for every graph G on n vertices with an even number of edges R(G, Z2) ≤ n + 2. This estimate is sharp.
Set systems with no union of cardinality 0 modulo m
"... Let q be a prime power. It is shown that for any hypergraph F = {F1,..., Fd(q−1)+1} whose maximal degree is d, there exists ∅ �= F0 ⊂ F, such that  � F F  ≡ 0 (mod q). ∈F0 For integers d, m ≥ 1 let fd(m) denote the minimal t such that for any hypergraph F = {F1,..., Ft} whose maximal degree is d, ..."
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Cited by 5 (3 self)
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Let q be a prime power. It is shown that for any hypergraph F = {F1,..., Fd(q−1)+1} whose maximal degree is d, there exists ∅ �= F0 ⊂ F, such that  � F F  ≡ 0 (mod q). ∈F0 For integers d, m ≥ 1 let fd(m) denote the minimal t such that for any hypergraph F = {F1,..., Ft} whose maximal degree is d, there exists ∅ � = F0 ⊂ F, such that  � F F  ≡ 0 (mod m). ∈F0 Here we determine fd(m) when m is a prime power, and remark on the general case. Example: Let Aij 1 ≤ i ≤ m − 1, 1 ≤ j ≤ d, be pairwise disjoint sets, each of cardinality m, and let {v1,..., vm−1} be disjoint from all the Aij ’s. Now F = {Aij ∪ {vi} : 1 ≤ i ≤ m − 1, 1 ≤ j ≤ d} satisfies F  = d(m − 1) but  � F ∈F0  � ≡ 0 (mod m) for any ∅ � = F0 ⊂ F. Hence fd(m) ≥ d(m − 1) + 1. Theorem 1: If q is a prime power then fd(q) = d(q − 1) + 1. Proof: Let F = {F1,..., Ft}, t = d(q − 1) + 1, be a hypergraph of degree ≤ d, and consider the polynomial: p(x1,..., xt) = �
A UNIFIED THEORY OF ZEROSUM PROBLEMS, SUBSET SUMS AND COVERS OF Z
, 2004
"... Zerosum problems on abelian groups, subset sums in a field and covers of the integers by residue classes, are three different active topics initiated by P. Erdős and investigated by many researchers. In an earlier ..."
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Cited by 3 (3 self)
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Zerosum problems on abelian groups, subset sums in a field and covers of the integers by residue classes, are three different active topics initiated by P. Erdős and investigated by many researchers. In an earlier
J. Aust. Math. Soc. 94 (2013), 268–275 doi:10.1017/S1446788712000547 CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS
, 2013
"... We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m). ..."
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We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m).