Restricted sumsets and a conjecture of Lev

LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev. This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.     

An extension of the Erdős–Ginzburg–Ziv Theorem to hypergraphs

An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a sequence S, subsequence S^′, and set T, |T∩S| denotes the number of terms x of S with xT, and |S| denotes the length of S, and SS^′ denotes the subsequence of S obtained by deleting all terms in S^′. We first prove the following two additive number theory results.(1) Let S be a finite sequence of elements from an abelian group G. If S has an n-set partition, A=A₁,…,A_n, such that

then there exists a subsequence S^′ of S, with length |S^′|≤max{|S|−n+1,2n}, and with an n-set partition, , such that . Furthermore, if ||A_i|−|A_j||≤1 for all i and j, or if |A_i|≥3 for all i, then .(2) Let S be a sequence of elements from a finite abelian group G of order m, and suppose there exist a,bG such that . If |S|≥2m−1, then there exists an m-term zero-sum subsequence S^′ of S with or .Let be a connected, finite m-uniform hypergraph, and be the least integer n such that for every 2-coloring (coloring with the elements of the cyclic group ) of the vertices of the complete m-uniform hypergraph , there exists a subhypergraph isomorphic to such that every edge in is monochromatic (such that for every edge e in the sum of the colors on e is zero). As a corollary to the above theorems, we show that if every subhypergraph of contains an edge with at least half of its vertices monovalent in , or if consists of two intersecting edges, then . This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when is a single edge.     

Extremal problems concerning Kneser graphs

Let and be two intersecting families of k-subsets of an n-element set. It is proven that | | ≤ (_k−1ⁿ⁻¹) + (_k−1ⁿ⁻¹) holds for , and equality holds only if there exist two points a, b such that {a, b} ∩ F ≠ for all F . For an example showing that in this case max | | = (1−o(1))(_kⁿ) is given. This disproves an old conjecture of Erdös [7]. In the second part we deal with several generalizations of Kneser's conjecture.     

Simultaneous similarity of matrices

In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT⁻¹, TBT⁻¹). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set ⁰_n,2,r,π M_n × M_n (M_n = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø₁,…,ø_s in the entries of A and B whose values are constant on all pairs similar in _n,2,r,π to (A, B). The values of the functions ø_i(A, B), I = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in _n,2,r,π. Let S_n be the subspace of complex symmetric matrices in M_n. For (A, B) ε S_n × S_n we consider the similarity class (TAT^t, TBT^t), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ε S_n × S_n.     

On extending Pollard’s theorem for <Emphasis Type="Italic">t</Emphasis>-representable sums

Let t ≥ 1, let A and B be finite, nonempty subsets of an abelian group G, and let $ A\mathop + \limits_i B $ A\mathop + \limits_i B denote all the elements c with at least i representations of the form c = a + b, with a ∈ A and b ∈ B. For |A|, |B| ≥ t, we show that either

Cycles in bipartite graphs

For k an integer, let G(a, b, k) denote a simple bipartite graph with bipartition (A, B) where |A| = a ≥ 2, |B| = b ≥ k ≥ 2, and each vertex of A has degree at least k. We prove two results concerning the existence of cycles in G(a, b, k).     

An inverse theorem for the restricted set addition in Abelian groups

Let A be a set of k5 elements of an Abelian group G in which the order of the smallest nonzero subgroup is larger than 2k−3. Then the number of different elements of G that can be written in the form a+a^′, where a,a^′A, a≠a^′, is at least 2k−3, as it has been shown in [Gy. Károlyi, The Erdős–Heilbronn problem in Abelian groups, Israel J. Math. 139 (2004) 349–359]. Here we prove that the bound is attained if and only if the elements of A form an arithmetic progression in G, thus completing the solution of a problem of Erdős and Heilbronn. The proof is based on the so-called ‘Combinatorial Nullstellensatz.’     

Embeddings of bipartite graphs

If G is a bipartite graph with bipartition A, B then let G_m,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a₁, …, a_m, each vertex b of B by an independent set b₁,…, b_n, and each edge ab of G by the complete bipartite graph with edges a_ib_j (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of G_m,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess-current graphs.” They include new genus embeddings for a large class of bipartite graphs.     

Group ramsey theory

A subset S of a group G is said to be a sum-free set if $\text{S ∩ (S + S) = ?}$. Such a set is maximal if for every sum-free set T ? G, we have |T| ? |S|. Here, we generalize this concept, defining a sum-free set S to be locally maximal if for every sum free set T such that S ? T ? G, we have S = T. Properties of locally maximal sum-free sets are studied and the sets are determined (up to isomorphism) for groups of small order.     

Algebras whose equivalence relations are congruences

It is proved that all the equivalence relations of a universal algebra A are its congruences if and only if either |A| ≤ 2 or every operation f of the signature is a constant (i.e., f(a ₁ , . . . , a _n ) = c for some c ∈ A and all the a ₁ , . . . , a _n ∈ A) or a projection (i.e., f(a ₁ , . . . , a _n ) = a _i for some i and all the a ₁ , . . . , a _n ∈ A). All the equivalence relations of a groupoid G are its right congruences if and only if either |G| ≤ 2 or every element a ∈ G is a right unit or a generalized right zero (i.e., x _a  = y _a for all x, y ∈ G). All the equivalence relations of a semigroup S are right congruences if and only if either |S| ≤ 2 or S can be represented as S = A∪B, where A is an inflation of a right zero semigroup, and B is the empty set or a left zero semigroup, and ab = a, ba = a ² for a ∈ A, b ∈ B. If G is a groupoid of 4 or more elements and all the equivalence relations of it are right or left congruences, then either all the equivalence relations of the groupoid G are left congruences, or all of them are right congruences. A similar assertion for semigroups is valid without the restriction on the number of elements.     

On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) with ξ(p × k) unknown. Denote the columns of X by X₍₁₎ ,…, X_(k) and set ψ⁽⁰⁾(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ⁽ⁱ⁾(X, X) = min[ψ⁽ⁱ⁻¹⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |] and Ψ⁽ⁱ⁾(S, X) = min[ψ⁽⁰⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ⁽⁰⁾(S, X) is dominated by each of Ψ⁽ⁱ⁾(S, X), i = 1,…,k and for every i, ψ⁽ⁱ⁾(S, X) is better than ψ⁽ⁱ⁻¹⁾(S, X). In particular, ψ^(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X₍₁₎X′₍₁₎|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X₍₁₎X′₍₁₎ + + X_(k)X′_(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.     

Laguerre Expansions of Gel′fand-Shilov Spaces

The subspaces G_α, G^β, and G^β_α (α, β ≥ 0)of Schwartz′ space S⁺ in (0, + ∞) are associated with the Hankel transform in the same way as the Gel′fand-Shilov spaces S_α, S^β, and S^β_α are associated with the Fourier transform. Indeed, if we consider the Hankel transform H_γ (γ < −1) defined by _γ(ƒ)(t) = ∫^∞₀ (xt)^−γ/2x^γJ_γ([formula]) ƒ(x) dx then _γ is an isomorphism from G_α, G^β, and G^β_α onto G^α, G_β, and G^α_β respectively. So. the spaces G^α_α are invariant for _γ. In this paper, we characterize the spaces G^α_α (α > 1) in terms of their Fourier-Laguerre coefficients. Also, we characterize the range of the Fourier-Laplace operator _D defined by _D(ƒ)(w) = ∫^∞₀ ƒ(t) e^{−(1/2)((1 + w)/(1 − w))t} for w D = {w : |w| ≤ 1} when it acts on the space G^α_α.     

Solution of the Cameron-Erdös problem for groups of prime order

A subset A of a group G is sum-free if a + b does not belong to A for any a, b ∈ A. Asymptotics of the number of sum-free sets in groups of prime order are proved.     

Coloring of distance graphs with intervals as distance sets

Let D be a set of positive integers. The distance graph G(Z,D) with distance set D is the graph with vertex set Z in which two vertices x,y are adjacent if and only if |x−y|D. The fractional chromatic number, the chromatic number, and the circular chromatic number of G(Z,D) for various D have been extensively studied recently. In this paper, we investigate the fractional chromatic number, the chromatic number, and the circular chromatic number of the distance graphs with the distance sets of the form D_m,[k,k^′]={1,2,…,m}−{k,k+1,…,k^′}, where m, k, and k^′ are natural numbers with m≥k^′≥k. In particular, we completely determine the chromatic number of G(Z,D_m,[2,k^′]) for arbitrary m, and k^′.     

The Inertia Set of Nonnegative Symmetric Sign Pattern with Zero Diagonal

The inertia set of a symmetric sign pattern A is the set i(A) = {i(B) | B = B ^T ∈ Q(A)}, where i(B) denotes the inertia of real symmetric matrix B, and Q(A) denotes the sign pattern class of A. In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern A in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns A with zero diagonal that require unique inertia.     

NON-EMPTY CROSS-2-INTERSECTING FAMILIES OF SUBSETS

Let Cdenote the set of all k-subests of an n-set.Assume Alohtain in Ca,and A lohtain in (A,B) is called a cross-2-intersecting family if |A B≥2 for and A∈A,B∈B.In this paper,the best upper bounds of the cardinalities for non-empty cross-2-intersecting familles of a-and b-subsets are obtained for some a and b,A new proof for a Frankl-Tokushige theorem[6] is also given.     

On an alternative functional equation related to the Cauchy equation

We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, bB,b0,M a positive integer; find all functionsf:G B such that for every (x, y) G ×G the Cauchy differencef(x+y)–f(x)–f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG.     

A Lower Bound for {a+b:aA, bB, and P(a,b)≠0}

Let A and B be two finite subsets of a field . In this paper, we provide a non-trivial lower bound for {a+b:aA, bB, and P(a,b)≠0} where P(x,y) [x,y].     

Packing two forests into a bipartite graph

For two integers a and b, we say that a bipartite graph G admits an (a, b)-bipartition if G has a bipartition (X, Y) such that |X| = a and |Y| = b. We say that two bipartite graphs G and H are compatible if, for some integers a and b, both G and H admit (a, b)-bipartitions. In this note, we prove that any two compatible trees of order n can be packed into a complete bipartite graph of order at most n + 1. We also provide a family of infinitely many pairs of compatible trees which cannot be packed into a complete bipartite graph of the same order. A theorem about packing two forests into a complete bipartite graph is derived from this result. © 1996 John Wiley & Sons, Inc.     

Two New Classes of Difference Families

A Z-cyclic triplewhist tournament for 4n+1 players, or briefly a TWh(4n+1), is equivalent to a n-set {(a_i, b_i, c_i, d_i) | i=1, …, n} of quadruples partitioning Z_4n+1−{0} with the property that ⁿ_i=1 {±(a_i−c_i), ±(b_i−d_i)}=ⁿ_i=1 {±(a_i−b_i), ±(c_i−d_i)}=ⁿ_i=1 {±(a_i−d_i), ±(b_i−c_i)}=Z_4n+1−{0}. The existence problem for Z-cyclic TWh(p)'s with p a prime has been solved for p1 (mod 16). I. Anderson et al. (1995, Discrete Math.138, 31–41) treated the case of p≡5 (mod 8) while Y. S. Liaw (1996, J. Combin. Des.4, 219–233) and G. McNay (1996, Utilitas Math.49, 191–201) treated the case of p≡9 (mod 16). In this paper, besides giving easier proofs of these authors' results, we solve the problem also for primes p≡1 (mod 16). The final result is the existence of a Z-cyclic TWh(v) for any v whose prime factors are all≡1 (mod 4) and distinct from 5, 13, and 17.