Erdős, Paul; Hajnal, András; Sós, Vera T.; Szemeredi, E. More results on Ramsey-Turán type problems. (English) Zbl 0526.05031 Combinatorica 3, 69-81 (1983). In [Combinat. Struct. Appl., Proc. Calgary Internat. Conf. Calgary 1969, 407-410 (1970; Zbl 0253.05145)] V.T.Sós raised a general scheme of new problems that can be considered as common generalizations of the problems treated in the classical results of Ramsey and Turán. This paper is a continuation of a sequence of papers on this subject.One of the main results is the following: Given \(k\geq 2\) and \(\varepsilon> 0\), let \(G_n\) be a sequence of graphs of order \(n\) size at least \((1/2)\left(\frac{3k-5}{3k-2}+\varepsilon\right)n^2\) edges such that the cardinality of the largest independent set in \(G_n\) is \(o(n)\). Let \(H\) be any graph of arboricity at most \(k\). Then there exists an \(n_0\) such that all \(G_n\) with \(n> n_0\) contain a copy of \(H\). This result is best possible in the case \(H=K_{2k}\). Reviewer: L.Lesniak Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 33 Documents MSC: 05C35 Extremal problems in graph theory 05C55 Generalized Ramsey theory 05C05 Trees Keywords:arboricity; sequence of graphs; largest independent set Citations:Zbl 0253.05145 PDFBibTeX XMLCite \textit{P. Erdős} et al., Combinatorica 3, 69--81 (1983; Zbl 0526.05031) Full Text: DOI References: [1] B. Bollobás andP. Erdos, On a Ramsey–Turán type problem.J. C. T. (B) 21 (1976), 166–168. · Zbl 0337.05134 [2] W. Brown andM. Simonovits, Preprint. [3] P. Erdos, Graph Theory and Probability,Canadian J. Math. Journal of Mathematics 11 (1959), 34–38. · Zbl 0084.39602 [4] P. Erdos andVera T. Sós, Some remarks on Ramsey’s and Turán’s theorem.Coll. Math. Soc. J. Bolyai,4.Comb. Theory and its Appl., North-Holland (1969), 395–404. [5] P. Erdos andVera T. Sós, Problems and results on Ramsey–Turán type theorems,Proc. West Coast Conf. on Combinatorics, Graph Th. and Computing, Humboldt State Univ. Arcata (1979), 17–23. [6] P. Erdos andVera T. Sós, Problems and results on Ramsey–Turán type theorems II,Studia Sci. Math. Hung. 14 (1979), 27–36. [7] P. Erdoos andA. H. Stone, On the structure of linear graphs,Bull. Amer. Math. Soc. 52 (1946), 1087–1091. · Zbl 0063.01277 [8] J. Gillis, Note on a property of measurable sets,J. London Math. Soc. 11 (1936), 139–141. · Zbl 0014.05501 [9] Vera T. Sós, On extremal problems in graph theory,Comb. structures and their appl. Proc. Calgary International Conference, Calgary (1969), 407–410. [10] E. Szemerédi, Graphs without complete quadrilaterals (in Hungarian),Mat. Lapok 23 (1973), 113–116. · Zbl 0277.05134 [11] E. Szemerédi, On sets containing nok elements in arithmetic progression,Acta Arithmetica 27 (1975), 199–245. · Zbl 0303.10056 [12] E. Szemerédi, Regular partitions of graphs,Coll. internat. C. N. R. S. 260 Probl. Combin. et Th. Graphes (1978), 399–402. · Zbl 0413.05055 [13] A. A. Zykov, On some properties of linear complexes (in Russian),Mat. Sbornik, N. S. 24 (1949), 163–188, andAmer. Math. Soc. Trans. 79 (1952). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.