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Let H be a graph on n vertices and let the blow-up graph G[H] be defined as follows. We replace each vertex v i of H by a cluster A i and connect some pairs of vertices of A i and A j if (v i , v j) was an edge of the graph H. As usual, we define the edge density between A i and A j as d(A i , A j) = e(A i , A j) |A i ||A j |. We study the following(More)
Starting with a result in combinatorial number theory we prove that (apart from a couple of exceptions that can be classified), for any elements a 1 ,. .. , a n of GF (q), there are distinct field elements b 1 ,. .. , b n such that a 1 b 1 + · · · + a n b n = 0. This implies the classification of hyperplanes lying in the union of the hyperplanes X i = X j(More)
In this paper we propose a multipartite version of the classical Turán problem of determining the minimum number of edges needed for an arbitrary graph to contain a given subgraph. As it turns out, here the non-trivial problem is the determination of the minimal edge density between two classes that implies the existence of a given subgraph. We determine(More)
We study the function M (n, k) which denotes the number of maximal k-uniform intersecting families F ⊆ [n] k. Improving a bound of Balogh, Das, Delcourt, Liu and Sharifzadeh on M (n, k), we determine the order of magnitude of log M (n, k) by proving that for any fixed k, M (n, k) = n Θ((2k k)) holds. Our proof is based on Tuza's set pair approach. The main(More)
The isolation of populations in the Iberian, Italian and Balkan peninsulas during the ice ages define four main paradigms that explain much of the known distribution of intraspecific genetic diversity in Europe. In this study we investigated the phylogeography of a wide-spread bat species, the bent-winged bat, Miniopterus schreibersii around the(More)
We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on n vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges |ERB| is given. The aim is to find the maximal size f of a monochromatic clique which is guaranteed by such a coloring.(More)