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On extremal problems of graphs and generalized graphs. (English) Zbl 0129.39905
An \(r\)-graph \(G\) consists of a set \(V(G)\) of elements called vertices of \(G\) and a set \(E(G)\) whose elements (called edges of \(G\)) are subsets of \(V(G)\) with cardinal number \(r\). (Thus a 2-graph is a graph in the usual sense.) The paper deals with the following problem: given positive integers \(n,r,l\), estimate the smallest value of \(f\) such that, for every \(r\)-graph \(G\) with \(n\) vertices and \(f\) edges, \(V(G)\) has \(r\) disjoint subsets \(S_1,...,S_r\) of cardinal number \(l\) such that \(\{x_1,...,x_r\} \in E(G)\) whenever \(x_1 \in S_1,...,x_r \in S_r\). Some related matters are also briefly discussed and some interesting results and unsolved problems in this area are mentioned.

MSC:
05C35 Extremal problems in graph theory
Keywords:
topology
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