Summary
Given a hereditary graph property \(\mathcal{P}\) let \({\mathcal{P}}^{n}\) be the set of those graphs in \(\mathcal{P}\) on the vertex set {1, …, n}. Define the constant c n by \(\vert {\mathcal{P}}^{n}\vert = {2}^{c_{n}\left ({ n \atop 2} \right )}\). We show that the limit lim n → ∞ c n always exists and equals 1 − 1 ∕ r, where r is a positive integer which can be described explicitly in terms of \(\mathcal{P}\). This result, obtained independently by Alekseev, extends considerably one of Erdős, Frankl and Rödl concerning principal monotone properties and one of Prömel and Steger concerning principal hereditary properties.
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References
V.E. Alekseev, Hereditary classes and coding of graphs, Probl. Cybern. 39 (1982) 151–164 (in Russian).
V.E. Alekseev, On the entropy values of hereditary classes of graphs, Discrete Math. Appl. 3 (1993) 191–199.
B. Bollobás, Extremal Graph Theory, Academic Press (1978), xx+488pp.
B. Bollobás, Hereditary and monotone properties of combinatorial structures, in Surveys in combinatorics 2007 (A. Hilton and J. Talbot, eds), London Mathematical Society Lecture Note Series 346, Cambridge University Press (2007), pp. 1–39.
B. Bollobás and P. Erdős, On the structure of edge graphs, Bull. London Math. Soc. 5 (1973) 317–321.
B. Bollobás, P. Erdős and M. Simonovits, On the structure of edge graphs II, J. London Math. Soc. 12(2) (1976) 219–224.
B. Bollobás and A. Thomason, Projections of bodies and hereditary properties of hypergraphs, J. London Math. Soc. 27 (1995) 417–424.
B. Bollobás and A. Thomason, The structure of hereditary properties and colourings of random graphs, Combinatorica 20 (2000) 173–202.
V. Chvátal and E. Szemerédi, On the Erdős-Stone theorem, J. London Math. Soc. 23 (1981) 207–214.
P. Erdős, P. Frankl and V. Rödl, The asymptotic enumeration of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs and Combinatorics 2 (1986), 113–121.
P. Erdős, D.J. Kleitman and B.L. Rothschild, Asymptotic enumeration of K n -free graphs, in International Coll. Comb., Atti dei Convegni Lincei (Rome) 17 (1976) 3–17.
P. Erdős and A.H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946) 1087–1091.
D.J. Kleitman and B.L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc. 205 (1975) (205–220).
Ph.G. Kolaitis, H.J. Prömel and B.L. Rothschild, K l + 1-free graphs: asymptotic structure and a 0-1-law, Trans. Amer. Math. Soc. 303 (1987) 637–671.
E. Marchant and A. Thomason, Extremal graphs and multigraphs with two weighted colours, in “Fete of Combinatorics and Computer Science” Bolyai Soc. Math. Stud., 20 (2010), 239–286.
E. Marchant and A. Thomason, The structure of hereditary properties and 2-coloured multigraphs, Combinatorica 31 (2011), 85–93.
H.J. Prömel and A. Steger, Excluding induced subgraphs: quadrilaterals, Random Structures and Algorithms 2 (1991) 55–71.
H.J. Prömel and A. Steger, Excluding induced subgraphs II: Extremal graphs, Discrete Applied Mathematics, 44 (1993) 283–294.
H.J. Prömel and A. Steger, Excluding induced subgraphs III: a general asymptotic, Random Structures and Algorithms 3 (1992) 19–31.
H.J. Prömel and A. Steger, The asymptotic structure of H-free graphs, in Graph Structure Theory (N. Robertson and P. Seymour, eds), Contemporary Mathematics 147, Amer. Math. Soc., Providence, 1993, pp. 167–178.
H.J. Prömel and A. Steger, Almost all Berge graphs are perfect, Combinatorics, Probability and Computing 1 (1992) 53–79.
F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30(2) (1929) 264–286.
V. Rödl (personal communication).
V. Rödl, Universality of graphs with uniformly distributed edges, Discrete Mathematics 59 (1986) 125–134.
D. Saxton and A. Thomason, Hypergraph containers (preprint).
E.R. Scheinerman and J. Zito, On the size of hereditary classes of graphs, J. Combinatorial Theory (B), 61 (1994) 16–39.
E. Szemerédi, Regular partitions of graphs, in Proc. Colloque Inter. CNRS (J.-C. Bermond, J.-C. Fournier, M. las Vergnas, D. Sotteau, eds), (1978).
A. Thomason, Graphs, colours, weights and hereditary properties, in ‘Surveys in Combinatorics, 2011 (R. Chapman ed.), LMS Lecture Note Series 392 (2011), 333–364.
P. Turán, On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok 48 (1941) 436–452.
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Bollobás, B., Thomason, A. (2013). Hereditary and Monotone Properties of Graphs. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős II. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7254-4_6
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