An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even: If both m and n are even, then every pair of sides may cross and so the answer is mn. If exactly one polygon, say the n-gon, has an odd number of sides, it can intersect each side of the m-gon polygon at most n−1 times; hence there are at most mn−m intersections. It is not hard to construct examples that meet these bounds. If both m and n are odd, the best known construction has mn−(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn−(m+⌈n/6⌉), for m≥n. We prove a new upper bound of mn−(m+n)+C for some constant C, which is optimal apart from the value of C

An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even: If both m and n are even, then every pair of sides may cross and so the answer is mn. If exactly one polygon, say the n-gon, has an odd number of sides, it can intersect each side of the m-gon polygon at most n−1 times; hence there are at most mn−m intersections. It is not hard to construct examples that meet these bounds. If both m and n are odd, the best known construction has mn−(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn−(m+⌈n/6⌉), for m≥n. We prove a new upper bound of mn−(m+n)+C for some constant C, which is optimal apart from the value of C

Convex hulls of random walks

We study the convex hulls of random walks establishing both law of large numbers and weak convergence statements for the perimeter length, diameter and shape of the hull. It should come as no surprise that the case where the random walk has drift, and the zero-drift case behave differently. We make use of several different methods to gain a better insight into each case. Classical results such as Cauchy’s surface area formula, the law of large numbers and the central limit theorem give some preliminary law of large number results. Considering the convergence of the random walk and then using the continuous mapping theorem leads to intuitive results in the case with drift where, under the appropriate scaling, non-zero, deterministic limits exist. In the zero-drift case the random limiting process, Brownian motion, provides insight into the behaviour of such a walk. We add to the literature in this area by establishing tighter bounds on the expected diameter of planar Brownian motion. The Brownian motion process is also useful for proving that the convex hull of the zero-drift random walk has no limiting shape. In the case with drift, a martingale difference method was used by Wade and Xu to prove a central limit theorem for the perimeter length. We use this framework to establish similar results for the diameter of the convex hull. Time-space processes give degenerate results here, so we use some geometric properties to further what is known about the variance of the functionals in this case and to prove a weak convergence statement for the diameter. During the study of the geometrical properties, we show that, only finitely often is there a single face in the convex minorant (or concave majorant) of such a walk

Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend

On the structure of graphs with forbidden induced substructures

One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints. In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs. Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every $2$-edge-colouring of the complete graph on $n$ vertices there is a monochromatic clique on at least $\frac{1}{2}\log n$ vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs. In the second part of this thesis we focus more on order-size pairs; an order-size pair $(n,e)$ is the family consisting of all graphs of order $n$ and size $e$, i.e. on $n$ vertices with $e$ edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs $(m,f)$, i.e. for $n$ approaching infinity, the limit superior of the fraction of all possible sizes $e$, such that the order-size pair $(n,e)$ does not avoid the pair $(m,f)$

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum