Related papers: On Kemnitz' Conjecture Concerning Lattice Points i…

For a finite set $A\subset \mathbb{R}^d$, let $\Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $\gamma_d(n)$ denote the largest…

We prove a constant term conjecture of Robbins and Zeilberger (J. Combin. Theory Ser. A 66 (1994), 17-27), by translating the problem into a determinant evaluation problem and evaluating the determinant. This determinant generalizes the…

A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the…

We study the statistical properties of the counting function of lattice points inside thin annuli. By a conjecture of Bleher and Lebowitz, if the width shrinks to zero, but the area converges to infinity, the distribution converges to the…

In 1998 Manickam and Singhi conjectured that for every positive integer $d$ and every $n \ge 4d$, every set of $n$ real numbers whose sum is nonnegative contains at least $\binom {n-1}{d-1}$ subsets of size $d$ whose sums are nonnegative.…

A famous theorem of Kirkman says that there exists a Steiner triple system of order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1973, Erd\H{o}s conjectured that one can find so-called `sparse' Steiner triple systems. Roughly speaking, the…

The Schinzel Hypothesis is a conjecture about irreducible polynomials in one variable over the integers: under some standard condition, they should assume infinitely many prime values at integers. We consider a relative version: if the…

In 1965 Erd\H os conjectured that for all $k\ge2$, $s\ge1$ and $n\ge k(s+1)$, an $n$-vertex $k$-uniform hypergraph $\F$ with $\nu(\F)=s$ cannot have more than \newline $\max\{\binom{sk+k-1}k,\;\binom nk-\binom{n-s}k\}$ edges. It took almost…

The big-line-big-clique conjecture states that for all $k,\ell\geq2$ there is an integer $n$ such that every finite set of at least $n$ points in the plane contains $\ell$ collinear points or $k$ pairwise visible points. We show that this…

This paper has two objectives. First, we study lattices with skew-Hermitian forms over division algebras with positive involutions. For division algebras of Albert types I and II, we show that such a lattice contains an "orthogonal" basis…

K. Adaricheva and M. Bolat have recently proved that if $U_0$ and $U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $U_{1-k}$ is included in the convex hull of…

Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P. This confirms, for large n, a conjecture of…

The classical Steinitz theorem asserts that if the origin lies within the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points in $S$ whose convex hull contains the origin within its interior.…

Let $P$ be a poset of size $2^k$ that has a greatest and a least element. We prove that, for sufficiently large $n$, the Boolean lattice $2^{[n]}$ can be partitioned into copies of $P$. This resolves a conjecture of Lonc.

Sierpinski's Hypothesis H1, formulated in 1958, is the conjecture that (provided $n\geq 2$), when the first $n^2$ counting numbers, $1, 2,3,\dots n^2$, are arranged in a square, then each row contains at least one prime. This conjecture is…

Let $L$ be any integral lattice in the 2-dimensional Euclidean space. Generalizing the earlier works of Hiroshi Maehara and others, we prove that for every integer $n>0$, there is a circle in the plane $\mathbb{R}^{2}$ that passes through…

Let $M$ be a submanifold of ${\Bbb P}^N$ of dimension $n>2$. Suppose that $(M,{\Cal O}_M(1))\cong{\Bbb P}({\Cal E}),{\Cal O}(1))$ for some vector bundle ${\Cal E}$ on a surface $S$. Then $N\ge 2n-1$ by Barth-Lefschetz Theorem. We are…

A collection $\mathcal{C}$ of $k$-element subsets of $\{1,2,\ldots,m\}$ is weakly separated if for each $I, J \in \mathcal{C}$, when the integers $1,2,\ldots,m$ are arranged around in a circle, there is a chord separating $I \backslash J$…

Let $d$ and $k$ be integers with $1 \leq k \leq d-1$. Let $\Lambda$ be a $d$-dimensional lattice and let $K$ be a $d$-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of $k$-dimensional…

We help Alice play a certain "convergence game" against Bob and win the prize, which is a constructive solution to a problem by Erd\H{o}s and Graham, posed in their 1980 book on open questions in combinatorial number theory. Namely, after…