DYNAMIC CLASSIFICATION OF GEOGRAPHIC POINTS ON GOOGLE MAPS

Classification of geographical points on Google maps is an interesting example of the use of cluster analysis algorithm in which the final number of clusters is obtained not only by presuppositions and the algorithm used, but also by the scale, on which map is actually displayed. The ultimate goal of classification is not only to obtain relatively homogeneous clusters, but also to prevent the phenomenon of "blurring" partitions on the map. In the paper there is proposed an algorithm that automatically creates a hierarchical structure of classes (which differs, however, from the structures obtained by the hierarchical agglomerative methods), in such way that the final classification takes into account the enlargement in which the map is displayed. The aim of article is illustrated with real examples on Google maps using JavaScript / JQuery

Monochromatic loose paths in multicolored kk-uniform cliques

For integers $k\ge 2$ and $\ell\ge 0$, a $k$-uniform hypergraph is called a loose path of length $\ell$, and denoted by $P_\ell^{(k)}$, if it consists of $\ell$ edges $e_1,\dots,e_\ell$ such that $|e_i\cap e_j|=1$ if $|i-j|=1$ and $e_i\cap e_j=\emptyset$ if $|i-j|\ge2$. In other words, each pair of consecutive edges intersects on a single vertex, while all other pairs are disjoint. Let $R(P_\ell^{(k)};r)$ be the minimum integer $n$ such that every $r$-edge-coloring of the complete $k$-uniform hypergraph $K_n^{(k)}$ yields a monochromatic copy of $P_\ell^{(k)}$. In this paper we are mostly interested in constructive upper bounds on $R(P_\ell^{(k)};r)$, meaning that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of $P_\ell^{(k)}$ in every coloring. In particular, we show that there is a constant $c>0$ such that for all $k\ge 2$, $\ell\ge3$, $2\le r\le k-1$, and $n\ge k(\ell+1)r(1+\ln(r))$, there is an algorithm such that for every $r$-edge-coloring of the edges of $K_n^{(k)}$, it finds a monochromatic copy of $P_\ell^{(k)}$ in time at most $cn^k$. We also prove a non-constructive upper bound $R(P_\ell^{(k)};r)\le(k-1)\ell r$

On the Ramsey-Tur\'an number with small ss-independence number

Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Tur\'an number $RT_s(n,H, f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with $\alpha_s(G) < f$, where $\alpha_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Tur\'an number attracted a considerable amount of attention and has been mainly studied for $f$ not too much smaller than $n$. In this paper we consider $RT_s(n,K_t, n^{\delta})$ for fixed $\delta<1$. We show that for an arbitrarily small $\varepsilon>0$ and $1/2<\delta< 1$, $RT_s(n,K_{s+1}, n^{\delta}) = \Omega(n^{1+\delta-\varepsilon})$ for all sufficiently large $s$. This is nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^{\delta}) = O(n^{1+\delta})$. Furthermore, the range of $\delta$ is as large as possible. We also consider more general cases and find bounds on $RT_s(n,K_{s+r},n^{\delta})$ for fixed $r\ge2$. Finally, we discuss a phase transition of $RT_s(n, K_{2s+1}, f)$ extending some recent result of Balogh, Hu and Simonovits.Comment: 25 p

Tight Hamilton Cycles in Random Uniform Hypergraphs

In this paper we show that $e/n$ is the sharp threshold for the existence of tight Hamilton cycles in random $k$-uniform hypergraphs, for all $k\ge 4$. When $k=3$ we show that $1/n$ is an asymptotic threshold. We also determine thresholds for the existence of other types of Hamilton cycles.Comment: 9 pages. Updated to add materia

The set chromatic number of random graphs

In this paper we study the set chromatic number of a random graph $G(n,p)$ for a wide range of $p=p(n)$. We show that the set chromatic number, as a function of $p$, forms an intriguing zigzag shape