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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)

In this paper, we study (1 : b) Avoider-Enforcer games played on the edge set of the complete graph on n vertices. For every constant k ≥ 3 we analyse the k-star game, where Avoider tries to avoid claiming k edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games – the strict and the monotone – and for each provide explicit… (More)

Generalizing a result in the theory of finite fields we prove that, apart from a couple of exceptions that can be classified, for any elements a 1 ,. .. , a m of the cyclic group of order m, there is a permutation π such that 1a π(1) + · · · + ma π(m) = 0.

To study how balanced or unbalanced a maximal intersecting family F ⊆ [n] r is we consider the ratio R(F) = ∆(F) δ(F) of its maximum and minimum degree. We determine the order of magnitude of the function m(n, r), the minimum possible value of R(F), and establish some lower and upper bounds on the function M (n, r), the maximum possible value of R(F). To… (More)

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