相关论文: Crescent configurations
Let $ES(n)$ denote the minimum natural number such that every set of $ES(n)$ points in general position in the plane contains $n$ points in convex position. In 1935, Erd\H{o}s and Szekeres proved that $ES(n) \le {2n-4 \choose n-2}+1$. In…
In this paper, we consider saturation problems related to the celebrated Erd\H{o}s--Szekeres convex polygon problem. For each $n \ge 7$, we construct a planar point set of size $(7/8) \cdot 2^{n-2}$ which is saturated for convex $n$-gons.…
Dorais asked for the maximum guaranteed size of a dimension $d$ subposet of an $n$-element poset. A lower bound of order $\sqrt{n}$ was found by Goodwillie. We provide a sublinear upper bound for each $d$. For $d=2$, our bound is…
In this paper, we first describe how we can arrange any bodies on Figure-Eight without collision in a dense subset of $[0,T]$ after showing that the self-intersections of Figure-Eight will not happen in this subset. Then it is reasonable…
If two closed Jordan curves in the plane have precisely one point in common, then it is called a {\em touching point}. All other intersection points are called {\em crossing points}. The main result of this paper is a Crossing Lemma for…
Consider l lines in P^2 such that no three lines meet in a point. Let X(l) denote all points of intersections of these l lines. We describe all pairs (d,l) such that generic degree d curve in P^2 contains a X(l).
In 2002, Koh and Tay conjectured that every bridgeless graph of order $n\geq 5$ and size at least ${n\choose 2}-n+5$ has an orientation of diameter two. Later, Cochran, Czabarka, Dankelmann and Sz\'{e}kely proved this conjecture and asked…
We prove a new upper bound for the minimum $d$-degree threshold for perfect matchings in $k$-uniform hypergraphs when $d<k/2$. As a consequence, this determines exact values of the threshold when $0.42k \le d < k/2$ or when $(k,d)=(12,5)$…
In 1980, Paul Erd\H{o}s posed the following problem: For every positive integer $n,$ determine a nonhamiltonian graph of order $n$ having the maximum number of Hamilton paths. We solve the more general problem of determining the…
The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most…
We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a…
Erd\H{o}s posed the question whether there exist infinitely many sets of consecutive numbers whose least common multiple (lcm) exceeds the lcm of another, larger set with greater consecutive numbers. In this paper, we answer this question…
In 1975, Erd\H{o}s asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erd\H{o}s observed that the upper bound $O(n^{5/3})$ holds since…
A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of…
A long-standing open conjecture of Branko Gr\"unbaum from 1972 states that any simple arrangement of $n$ pairwise intersecting pseudocircles in the plane can have at most $2n-2$ digons. Agarwal et al. proved this conjecture for arrangements…
We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the…
Multiple-choice load balancing has been a topic of intense study since the seminal paper of Azar, Broder, Karlin, and Upfal. Questions in this area can be phrased in terms of orientations of a graph, or more generally a k-uniform random…
For a positive integer $d$, a set of points in $d$-dimensional Euclidean space is called almost-equidistant if for any three points from the set, some two are at unit distance. Let $f(d)$ denote the largest size of an almost-equidistant set…
Suppose that $nk$ points in general position in the plane are colored red and blue, with at least $n$ points of each color. We show that then there exist $n$ pairwise disjoint convex sets, each of them containing $k$ of the points, and each…
A rational distance set in the plane is a point set which has the property that all pairwise distances between its points are rational. Erd\H os and Ulam conjectured in 1945 that there is no dense rational distance set in the plane. In this…