Simonovits, Miklós

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Extremal graph theory, Stability, andAnti-Ramsey theoremsMiklo´s SimonovitsAlfre´d Re´nyi Mathematical Instituteof the Hungarian Academy of Sciences,BudapestExtremal graph theory is one of the most developed branches of DiscreteMathematics. Stability methods are very successful in this field. We shall givesome illustration of this method for graphs, hypergraphs, and Anti-Ramseyproblems. An Anti-Ramsey problem is where a sample graph L is fixed and wecolour the edges of , e.g., a complete graph Kn without having a copy of L inwhich all the edges have distinct colours.We shall consider Dual Anti-Ramsey problems, coming from TheoreticalComputer Science. Burr, Erdo˝s, Graham and T. So´s [1] defined and investigateda dual variant of the Anti-Ramsey problems.The dual Anti-Ramsey problem. Fix a sample graph L, andconsider a (variable) graph Gn on n vertices, with e = e(Gn) >ex(n,L) edges. Let χS(Gn, L) denote the minimum number ofcolours needed to colour the edges of Gn so that no L ⊆ Gn hastwo edges of the same colour. DetermineχS(n, e, L) := min {χS(Gn, L) : e(Gn) = e} .Here we improve several results of [1] and [2].This lecture is partly based on a manuscript of Erdo˝s and Simonovits [3]from the late 1980’s.References[1] S. A. Burr, P. Erdo˝s, R. L. Graham and Vera T. So´s: Maximal antiramsey graphsand the strong chromatic number, Journal of Graph Theory, vol 13(3), (1989) pp163–182.[2] S. Burr, P. Erdo˝s, P. Frankl, R. L. Graham, V. T. So´s: Further results on maximalAnti–Ramsey graphs Proc. Kalamazoo Combin. Conf., 1989 pp 193–206[3] P. Erdo˝s and M. Simonovits: How many colours are needed to colour everypentagon of a graph in five colours? (manuscript, under publication)1

Extremal Graph Theory, Stability, and Anti-Ramsey Theorems

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Extremal Graph Theory, Stability, and Anti-Ramsey Theorems

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