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 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.     

An Erdős-Ko-Rado theorem for finite classical polar spaces

Consider a finite classical polar space of rank \(d\ge 2\) and an integer n with \(0<n<d\). In this paper, it is proved that the set consisting of all subspaces of rank n that contain a given point is a largest Erd?s-Ko-Rado set of subspaces of rank n of the polar space. We also show that there are no other Erd?s-Ko-Rado sets of subspaces of rank n of the same size.     

A Weighted Erdős-Ginzburg-Ziv Theorem

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 so that they can be considered as sets. If S is a sequence of m+n−1 elements from a finite abelian group G of order m and exponent k, and if is a sequence of integers whose sum is zero modulo k, then there exists a rearranged subsequence of S such that . This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when m = n and w_i = 1 for all i, and confirms a conjecture of Y. Caro. Furthermore, we in part verify a related conjecture of Y. Hamidoune, by showing that if S has an n-set partition A=A₁, . . .,A_n such that |w_iA_i| = |A_i| for all i, then there exists a nontrivial subgroup H of G and an n-set partition A′ =A′₁, . . .,A′_n of S such that and for all i, where w_iA_i={w_ia_i |a_i∈A_i}.     

The Erdös-Kac theorem for additive arithmetical semigroups

We show that the Erdös-Kac theorem for additive arithmetical semigroups can be proved under the condition that the counting function of elements has the asymptotics G(n) = q ⁿ (A + O(1/(lnn)^k) as n → ∞ with A > 0, q > 1, and arbitrary k ∈ ? and that P(n) = O(q ⁿ /n) for the number of prime elements of degree n. This improves a result of Zhang.     

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.     

An Erdős–Fuchs theorem for ordered representation functions

The Ramanujan Journal - Let $$k\ge 2$$ be a positive integer. We study concentration results for the ordered representation functions $$r^{{ \le }}_k({\mathcal {A}},n) = \# \big \{ (a_1 \le \dots...     

An extension of the Erdős-Stone theorem

LetK _p(u₁, ..., u_p) be the completep-partite graph whoseith vertex class hasu _i vertices (lip). We show that the theorem of Erds and Stone can be extended as follows. There is an absolute constant >0 such that, for allr1, 0<1 and=">1/r, every graphG=G ⁿ of sufficiently large order |G|=n with at least

A stability version for a theorem of Erdős on nonhamiltonian graphs

Let $n,d$ be integers with $1\le d\le ?\frac{n?1}{2}?$, and set $h(n,d)?\left(\genfrac{}{}{0ex}{}{n?d}{2}\right)+d^2$ and $e(n,d)?max\{h(n,d),h(n,?\frac{n?1}{2}?)\}$. Because $h(n,d)$ is quadratic in $d$, there exists a $d_0\left(n\right)=(n∕6)+O\left(1\right)$ such that

$e(n,1)>e(n,2)>?>e(n,d_0)=e(n,d_0+1)=?=e\left(n,?\frac{n?1}{2}?\right).$

A theorem by Erd?s states that for $d\le ?\frac{n?1}{2}?$, any $n$-vertex nonhamiltonian graph $G$ with minimum degree $\delta \left(G\right)\ge d$ has at most $e(n,d)$ edges, and for $d>d_0\left(n\right)$ the unique sharpness example is simply the graph $K_n?E\left(K_{?(n+1)∕2?}\right)$. Erd?s also presented a sharpness example $H_{n,d}$ for each $1\le d\le d_0\left(n\right)$.We show that if $d<d_0\left(n\right)$ and a $2$-connected, nonhamiltonian $n$-vertex graph $G$ with $\delta \left(G\right)\ge d$ has more than $e(n,d+1)$ edges, then $G$ is a subgraph of $H_{n,d}$. Note that $e(n,d)?e(n,d+1)=n?3d?2\ge n∕2$ whenever $d<d_0\left(n\right)?1$.     

On the Erd?s-Ginzburg-Ziv constant of finite abelian groups of high rank

Let G be a finite abelian group. The Erd?s-Ginzburg-Ziv constant s(G) of G is defined as the smallest integer l∈N such that every sequence S over G of length |S|?l has a zero-sum subsequence T of length |T|=exp(G). If G has rank at most two, then the precise value of s(G) is known (for cyclic groups this is the theorem of Erd?s-Ginzburg-Ziv). Only very little is known for groups of higher rank. In the present paper, we focus on groups of the form , with n,r∈N and n?2, and we tackle the study of s(G) with a new approach, combining the direct problem with the associated inverse problem.     

Clarkson-Erdős theorem for many variables

In the present paper, the Clarkson-Erdős, which complements the well-known Müntz theorem, is extended to the case of many variables. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 594–598, April, 1999.     

Erdős–Szekeres Theorem for Lines

According to the Erd?s–Szekeres theorem, for every n, a sufficiently large set of points in general position in the plane contains n in convex position. In this note we investigate the line version of this result, that is, we want to find n lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erd?s–Szekeres theorem.     

Erdős-Szekeres Tableaux

We explore a question related to the celebrated Erd?s-Szekeres Theorem and develop a geometric approach to answer it. Our main object of study is the Erd?s-Szekeres Tableau, or EST, of a number sequence. An EST is the sequence of integral points whose coordinates record the length of the longest increasing and longest decreasing subsequence ending at each element of the sequence. We define the Order Poset of an EST in order to answer the question: What information about the sequence can be recovered by its EST?     

A discrete isodiametric result: The Erdős–Ko–Rado theorem for multisets

There are many generalizations of the Erdős–Ko–Rado theorem. Here the new results (and problems) concern families of $t$-intersecting $k$-element multisets of an $n$-set. We point out connections to coding theory and geometry. We verify the conjecture that for $n\ge t(k-t)+2$ such a family can have at most $\left(\genfrac{}{}{00}{}{n+k-t-1}{k-t}\right)$ members.