Abstract
We shall consider graphs (hypergraphs) without loops and multiple edges. Let ℒ be a family of so called prohibited graphs and ex (n, ℒ) denote the maximum number of edges (hyperedges) a graph (hypergraph) onn vertices can have without containing subgraphs from ℒ. A graph (hyper-graph) will be called supersaturated if it has more edges than ex (n, ℒ). IfG hasn vertices and ex (n, ℒ)+k edges (hyperedges), then it always contains prohibited subgraphs. The basic question investigated here is: At least how many copies ofL ε ℒ must occur in a graphG n onn vertices with ex (n, ℒ)+k edges (hyperedges)?
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References
B. Bollobás,Extremal graph theory, London Math. Soc. Monographs, No.11 Academic Press, 1978.
B. Bollobás. Relations between sets of complete subgraphs,Proc. Fifth British Comb. Conference, Aberdeen, 1975, 79–84.
B. Bollobás, On complete subgraphs of different orders,Math. Proc. Cambridge Phil. Soc. 79 (1976), 19–24.
B. Bollobás, P. Erdős andM. Simonovits. On the structure of edge graphs II,J. London Math. Soc. 12 (2) (1976) 219–224.
P. Erdős, On a theorem of Rademacher—Turán,Illinois J. Math. 6 (1962) 122–127.
P. Erdős, On the number of complete subgraphs contained in certain graphs,Publ. Math. Inst. Hungar. Acad. Sci. 7 (1962) 459–464.
P. Erdős, On extremal problems of graphs and generalized graphs,Israel J. Math. 2 (1965) 183–190.
P. Erdős, Some recent results on extremal problems in graph theory,Theory of Graphs, Intern. Symp. Rome, 1966, 118–123.
P. Erdős. On some new inequalities concerning extremal properties of graphs,Theory of Graphs, Proc. Coll. Tihany, 1966, 77–81.
P. Erdős, On the number of complete subgraphs and circuits contained in graphs,Casopis Pest. Mat. 94 (1969) 290–296.
P. Erdős, On some extremal problems onr-graphs,Discrete Math. 1 (1971) 1–6.
P. Erdős andM. Simonovits, Compactness results in extremal graph theory,Combinatorica 2 (3) (1982), 275–288.
P. Erdős andM. Simonovits, A limit theorem in graph theory,Studia Sci. Math. Hungar. 1 (1966) 51–57.
P. Erdős andM. Simonovits, Some extremal problems in graph theory,Coll. Math. Soc. János Bolyai 4 (1969) 377–390.
P. Erdős andA. H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc.52 (1946) 1089–1091.
Gy. Katona, T. Nemetz andM. Simonovits, A new proof on a theorem of Turán and some remarks on a generalization of it, (in Hungarian)Matematikai Lapok 15 (1964) 228–238.
T. Kővári, V. T. Sós andP. Turán, On a problem of Zarankievicz,Coll. Math. 3 (1954) 50–57.
L. Lovász andM. Simonovits, On the number of complete subgraphs of a graph,Proc. Fifth British Combinatorial Coll., Aberdeen, (1975) 431–442.
L. Lovász andM. Simonovits, On the number of complete subgraphs of a graph, II.,Studies in Pure Mathematics, (1983), 459–495.
M. Simonovits, A method for solving extremal problems in graph theory,in: Theory of Graphs, Proc. Coll. Tihany, Hungary, 1966, 279–319.
P. Turán, On an extremal problem in graph theory, Mat. Fiz. Lapok48 (1941) 436–452.
G. R. Blakley andP. Roy,Proc. Amer. Math. Soc. 16 (1965) 1244–1245; see alsoH. P. Mulfiolland andC. A. B. Smith,American Mathematical Monthly 66 (1959) 673–683, andD. London,Duke Math. Journal 33 (1966) 511–522.