Erdős-Ginzburg-Ziv theorem for finite commutative semigroups

Let \(\mathcal{S}\) be a finite additively written commutative semigroup, and let \(\exp(\mathcal{S})\) be its exponent which is defined as the least common multiple of all periods of the elements in \(\mathcal{S}\) . For every sequence T of elements in \(\mathcal{S}\) (repetition allowed), let \(\sigma(T) \in\mathcal{S}\) denote the sum of all terms of T. Define the Davenport constant \(\mathsf{D}(\mathcal{S})\) of \(\mathcal{S}\) to be the least positive integer d such that every sequence T over \(\mathcal{S}\) of length at least d contains a proper subsequence T′ with σ(T′)=σ(T), and define \(\mathsf{E}(\mathcal{S})\) to be the least positive integer ? such that every sequence T over \(\mathcal{S}\) of length at least ? contains a subsequence T′ with \(|T|-|T'|= \lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil \exp(\mathcal{S})\) and σ(T′)=σ(T). When \(\mathcal{S}\) is a finite abelian group, it is well known that \(\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil\exp (\mathcal{S})=|\mathcal{S}|\) and \(\mathsf{E}(\mathcal{S})=\mathsf{D}(\mathcal{S})+|\mathcal{S}|-1\) . In this paper we investigate whether \(\mathsf{E}(\mathcal{S})\leq \mathsf{D}(\mathcal{S})+ \lceil\frac{|\mathcal{S}|}{\exp(\mathcal {S})} \rceil \exp(\mathcal{S})-1\) holds true for all finite commutative semigroups \(\mathcal{S}\) . We provide a positive answer to the question above for some classes of finite commutative semigroups, including group-free semigroups, elementary semigroups, and archimedean semigroups with certain constraints.     

The Erdös-Ko-Rado theorem for twisted Grassmann graphs

We present a “modern” approach to the Erdös-Ko-Rado theorem for Q-polynomial distance-regular graphs and apply it to the twisted Grassmann graphs discovered in 2005 by Van Dam and Koolen.     

Prime analogues of the Erd?s-Kac theorem for elliptic curves

Let E/Q be an elliptic curve. For a prime p of good reduction, let E(Fp) be the set of rational points defined over the finite field Fp. We denote by ω(#E(Fp)), the number of distinct prime divisors of #E(Fp). We prove that the quantity (assuming the GRH if E is non-CM)

     

New Erdös-Kac type theorems

Assuming a quasi Generalized Riemann Hypothesis (quasi-GRH for short) for Dedekind zeta functions over Kummer fields of the type we prove the following prime analogue of a conjecture of Erd?s & Pomerance (1985) concerning the exponent function f_a(p) (defined to be the minimal exponent e for which a^e ≡ 1 modulo p):

The exact bound in the Erdös-Ko-Rado theorem

This paper contains a proof of the following result: ifn≧(t+1)(k?t?1), then any family ofk-subsets of ann-set with the property that any two of the subsets meet in at leastt points contains at most \(\left( {\begin{array}{*{20}c} {n - t} \\ {k - t} \\ \end{array} } \right)\) subsets. (By a theorem of P. Frankl, this was known whent≧15.) The bound (t+1)(k-t-1) represents the best possible strengthening of the original 1961 theorem of Erdös, Ko, and Rado which reaches the same conclusion under the hypothesisn≧t+(k?t) \(\left( {\begin{array}{*{20}c} k \\ t \\ \end{array} } \right)^3 \) . Our proof is linear algebraic in nature; it may be considered as an application of Delsarte’s linear programming bound, but somewhat lengthy calculations are required to reach the stated result. (A. Schrijver has previously noticed the relevance of these methods.) Our exposition is self-contained.     

An Erdös-Ko-Rado theorem for restricted signed sets

A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.     

The Erdős-Ko-Rado theorem for finite affine spaces

In [W.N. Hsieh, Intersection theorems for finite vector spaces, Discrete Math. 12 (1975) 1–16], Hsieh obtained the Erd?s-Ko-Rado theorem for finite vector spaces. This paper generalizes Hsieh’s result and obtains the Erd?s-Ko-Rado theorem for finite affine spaces.     

The Erd?s-Ginzburg-Ziv theorem for finite solvable groups

Let G be a non-cyclic finite solvable group of order n, and let S=(g₁,…,g_k) be a sequence of k elements (repetition allowed) in G. In this paper we prove that if , then there exist some distinct indices i₁,i₂,…,i_n such that the product g_i1g_i2?g_in=1. This result substantially improves the Erd?s-Ginzburg-Ziv theorem and other existing results.     

The de Bruijn–Erd?s theorem for hypergraphs

Fix integers n ≥ r ≥ 2. A clique partition of ${{[n] \choose r}}$ is a collection of proper subsets ${A_1, A_2, \ldots, A_t \subset [n]}$ such that ${\bigcup_i{A_i \choose r}}$ is a partition of ${{[n]\choose r}}$ . Let cp(n, r) denote the minimum size of a clique partition of ${{[n] \choose r}}$ . A classical theorem of de Bruijn and Erd?s states that cp(n, 2) =?n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3, $${\rm cp}(n, r) \geq (1 + o(1))n^{r/2} \quad \quad {\rm as} \, \, n \rightarrow \infty.$$ We conjecture cp(n, r) =?(1 +?o(1))n ^r/2. This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman–Mullin–Stinson. We give further evidence of this conjecture by constructing, for each r ≥ 4, a family of (1?+?o(1))n ^r/2 subsets of [n] with the following property: no two r-sets of [n] are covered more than once and all but o(n ^r ) of the r-sets of [n] are covered. We also give an absolute lower bound ${{\rm cp}(n, r) \geq {n \choose r}/{q + r - 1 \choose r}}$ when n =?q ² + q +?r ? 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.     

An Erdös-Ko-Rado theorem for the subcubes of a cube

LetP be that partially ordered set whose elements are vectors x=(x ₁, ...,x _n ) withx _i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx _i =y _i orx _i =0 for alli. LetN _i (P)={x εP : |{j:x _j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN ₁(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f ₀, ...,f _n ) for which there is an Erdös-Ko-Rado familyF inP such that |N _i (P) ∩F|=f _i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.     

The Erdős-Szekeres theorem and congruences

The following problem of combinatorial geometry is considered. Given positive integers n and q, find or estimate a minimal number h for which any set of h points in general position in the plane contains n vertices of a convex polygon for which the number of interior points is divisible by q. For a wide range of parameters, the existing bound for h is dramatically improved.     

The Lidskii-Mirsky-Wielandt theorem – additive and multiplicative versions

Summary. We use a simple matrix splitting technique to give an elementary new proof of the Lidskii-Mirsky-Wielandt Theorem and to obtain a multiplicative analog of the Lidskii-Mirsky-Wielandt Theorem, which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of Hermitian matrices and singular values of general matrices. We apply our bound to obtain numerous bounds on the matching distance between the eigenvalues and singular values of matrices. Our results strengthen and generalize those in the literature. Received November 20, 1996 / Revised version received January 27, 1998     

The Gearhart-Prüss theorem for a class of degenerate semigroups of operators

We consider a class of semigroups of operators in Hilbert space whose generators are linear relations. An analog of the Gearhart-Prüss theorem for semigroups of operators from the class under consideration is obtained.     

A dual form of Erdös-Rado's canonization theorem

A dual form of the Erdös-Rado canonization theorem (J. London Math. Soc.25 (1950), 249–255) is established. We give several applications.     

An Erd?s-Ko-Rado theorem for partial permutations

Let [n] denote the set of positive integers {1,2,…,n}. An r-partial permutation of [n] is a pair (A,f) where A⊆[n], |A|=r and f:A→[n] is an injective map. A set A of r-partial permutations is intersecting if for any (A,f), (B,g)∈A, there exists x∈A∩B such that f(x)=g(x). We prove that for any intersecting family A of r-partial permutations, we have .It seems rather hard to characterize the case of equality. For 8?r?n-3, we show that equality holds if and only if there exist x₀ and ε₀ such that A consists of all (A,f) for which x₀∈A and f(x₀)=ε₀.