Critical hypergraphs and intersecting set-pair systems

Zsolt Tuza
1985 Journal of combinatorial theory. Series B (Print)  
A method is given for solving extremal problems of the following type: determine the maximal number of vertices in a class of hypergraphs. Results are applied for T-critical and v-critical hypergraphs. 0 1985 Academic Press, Inc. (i) Ai"Bj=(2/ if and only if i=j whenever 1 <ii, j<m. An ISP-system is called (a, @-system if, additionally, (ii) [Ai/ = a and IBil = b for every i < m. The heart of our method is Lemma 4 which says that a suitable (a, b)system can be gained from the hypergraph if a
more » ... dition holds. In this way an upper bound is obtained for the number of vertices, as the pairs of an (a, b)-system cannot cover too many points (Theorem 6). In the second part of the paper we apply this technique for various problems. In Section 3 we deal with v-critical hypergraphs of rank r. We improve the order of magnitude in a result of Lovasz [ 19, Theorem 10). So we gain the best possible estimate, apart from a constant factor, in case v = 1
doi:10.1016/0095-8956(85)90043-7 fatcat:5q3iozxw25agdcpgzvf4nshblm