Kemnitz’ conjecture revisited

AbstractA conjecture of Kemnitz remained open for some 20 years: each sequence of 4n-3 lattice points in the plane has a subsequence of length n whose centroid is a lattice point. It was solved independently by Reiher and di Fiore in the autumn of 2003. A refined and more general version of Kemnitz’ conjecture is proved in this note. The main result is about sequences of lengths between 3p-2 and 4p-3 in the additive group of integer pairs modulo p, for the essential case of an odd prime p. We derive structural information related to their zero sums, implying a variant of the original conjecture for each of the lengths mentioned. The approach is combinatorial

On the minimum diameter of plane integral point sets

Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter $d(2,n)$ of a plane integral point set consisting of $n$ points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for $d(2,n)$ achieving the known upper bound $n^{c_2\log\log n}$ up to a constant in the exponent. A famous question of Erd\H{o}s asks for plane integral point sets with no 3 points on a line and no 4 points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by $\dot{d}(2,n)$. Recently $\dot{d}(2,7)=22 270$ could be determined via an exhaustive search.Comment: 12 pages, 5 figure

On weighted zero-sum sequences

Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest positive integer $m$, denoted by s_A(G), such that any sequence {c_i}_{i=1}^m with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which there are a_1,...,a_n in A such that sum_{j=1}^na_ic_{i_j}=0. When G is a p-group, A contains no multiples of p and any two distinct elements of A are incongruent mod p, we show that s_A(G) is at most $\lceil D(G)/|A|\rceil+exp(G)-1$ if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is the Davenport constant of G and this upper bound for s_A(G)in terms of |A| is essentially best possible. In the case A={1,-1}, we determine the asymptotic behavior of s_{{1,-1}}(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank, s_{{1,-1}}(G)=exp(G)+log_2|G|+O(log_2log_2|G|) as exp(G) tends to the infinity. Combined with a lower bound of $exp(G)+sum{i=1}{r}\lfloor\log_2 n_i\rfloor$, where $G=\Z_{n_1}\oplus...\oplus \Z_{n_r}$ with 1<n_1|... |n_r, this determines s_{{1,-1}}(G), for even exponent groups, up to a small order error term. Our method makes use of the theory of L-intersecting set systems. Some additional more specific values and results related to s_{{1,-1}}(G) are also computed.Comment: 24 pages. Accepted version for publication in Adv. in Appl. Mat

Arithmetic-Progression-Weighted Subsequence Sums

Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\},$$ which is a particular kind of weighted restricted sumset. We show that $|W\odot S|\geq \min\{|G|-1,\,n\}$, that $W\odot S=G$ if $n\geq |G|+1$, and also characterize all sequences $S$ of length $|G|$ with $W\odot S\neq G$. This result then allows us to characterize when a linear equation $$a_1x_1+...+a_rx_r\equiv \alpha\mod n,$$ where $\alpha,a_1,..., a_r\in \Z$ are given, has a solution $(x_1,...,x_r)\in \Z^r$ modulo $n$ with all $x_i$ distinct modulo $n$. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G\cong C_{n_1}\oplus C_{n_2}$ (where $n_1\mid n_2$ and $n_2\geq 3$) having $k$ distinct terms, for any $k\in [3,\min\{n_1+1,\,\exp(G)\}]$. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs