Point Configurations in d -Space without Large Subsetsin Convex Position

In this paper we give a lower bound for the Erd\H os-Szekeres number in higher dimensions. Namely, in two different ways we construct, for every $n>d\ge 2$, a configuration of $n$ points in general position in $\R^d$ containing at most $c_d(\log n)^{d-1}$ points in convex position. (Points in $\R^d$ are in convex position if none of them lies in the convex hull of the others.

On the Erd\H{o}s-Tuza-Valtr Conjecture

The Erd\H{o}s-Szekeres conjecture states that any set of more than $2^{n-2}$ points in the plane with no three on a line contains the vertices of a convex $n$-gon. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that any set of more than $\sum_{i = n - b}^{a - 2} \binom{n - 2}{i}$ points in a plane either contains the vertices of a convex $n$-gon, $a$ points lying on a concave downward curve, or $b$ points lying on a concave upward curve. They also showed that the generalization is actually equivalent to the Erd\H{o}s-Szekeres conjecture. We prove the first new case of the Erd\H{o}s-Tuza-Valtr conjecture since the original 1935 paper of Erd\H{o}s and Szekeres. Namely, we show that any set of $\binom{n-1}{2} + 2$ points in the plane with no three points on a line and no two points sharing the same $x$-coordinate either contains 4 points lying on a concave downward curve or the vertices of a convex $n$-gon.Comment: 16 pages, 8 figure

Métrologie des graphes de terrain, application à la construction de ressources lexicales et à la recherche d'information

This thesis is organized in two parts : the first part focuses on measures of similarity (or proximity) between vertices of a graph, the second part on clustering methods for bipartite graph. A new measure of similarity between vertices, based on short time random walks, is introduced. The main advantage of the method is that it is insensitive to the density of the graph. A broad state of the art of similarities between vertices is then proposed, as well as experimental comparisons of these measures. This is followed by the proposal of a robust method for comparing graphs sharing the same set of vertices. This measure is shown to be applicable to the comparison and merging of synonymy networks. Finally an application for the enrichment of lexical resources is presented. It consists in providing candidate synonyms on the basis of already existing links. In the second part, a parallel between formal concept analysis and clustering of bipartite graph is established. This parallel leads to the particular case where a partition of one of the vertex groups can be determined whereas there is no corresponding partition on the other group of vertices. A simple method that addresses this problem is proposed and evaluated. Finally, a system of automatic classification of search results (Kodex) is presented. This system is an application of previously seen clustering methods. An evaluation on a collection of two million web pages shows the benefits of the approach and also helps to understand some differences between clustering methods.Cette thèse s'organise en deux parties : une première partie s'intéresse aux mesures de similarité (ou de proximité) définies entre les sommets d'un graphe, une seconde aux méthodes de clustering de graphe biparti. Une nouvelle mesure de similarité entre sommets basée sur des marches aléatoires en temps courts est introduite. Cette méthode a l'avantage, en particulier, d'être insensible à la densité du graphe. Il est ensuite proposé un large état de l'art des similarités entre sommets, ainsi qu'une comparaison expérimentale de ces différentes mesures. Cette première partie se poursuit par la proposition d'une méthode robuste de comparaison de graphes partageant le même ensemble de sommets. Cette méthode est mise en application pour comparer et fusionner des graphes de synonymie. Enfin une application d'aide à la construction de ressources lexicales est présentée. Elle consiste à proposer de nouvelles relations de synonymie à partir de l'ensemble des relations de synonymie déjà existantes. Dans une seconde partie, un parallèle entre l'analyse formelle de concepts et le clustering de graphe biparti est établi. Ce parallèle conduit à l'étude d'un cas particulier pour lequel une partition d'un des groupes de sommets d'un graphe biparti peut-être déterminée alors qu'il n'existe pas de partitionnement correspondant sur l'autre type de sommets. Une méthode simple qui répond à ce problème est proposée et évaluée. Enfin Kodex, un système de classification automatique des résultats d'une recherche d'information est présenté. Ce système est une application en RI des méthodes de clustering vues précédemment. Une évaluation sur une collection de deux millions de pages web montre les avantages de l'approche et permet en outre de mieux comprendre certaines différences entre méthodes de clustering

Ramsey-remainder

AbstractWe investigate the following Ramsey-type problem. Given a natural numberk,determine the smallest integerrr(k)such that, ifnis sufficiently large with respect tok,andSis any set ofnpoints in general position in the plane, then all but at mostrr(k)points ofScan be partitioned into convex sets of sizes ⩾k.We provide estimates onrr(k)which are best possible if a classic conjecture of Erdos and Szekeres on the Ramsey number for convex sets is valid. We also prove that in several types of combinatorial structures, the corresponding ‘Ramsey-remainder’rr(k)is equal to the off-diagonal Ramsey numberr(k,k-1)minus 1

Ramsey-remainder for convex sets and the Erdos-Szekeres theorem

AbstractAs a consequence of the Erdős–Szekeres theorem we prove that, for n large enough, any set of kn points, in general position in Ed,d⩾3, can be partitioned into n convex subsets of size k. Although this is far from being true for d=2, we find the exact conditions under which, for sufficiently large n, any set of 4n points, in general position in the plane, can be partitioned into n convex quadrilaterals. Moreover, we design an efficient algorithm which either finds such a partition, or indicates that such a partition does not exist, thus answering a question of Joe Mitchell