Chapter

The Mathematics of Paul Erdös II

Volume 14 of the series Algorithms and Combinatorics pp 70-78

Hereditary and Monotone Properties of Graphs

  • Béla BollobásAffiliated withDepartment of Pure Mathematics and Mathematical Statistics
  • , Andrew ThomasonAffiliated withDepartment of Pure Mathematics and Mathematical Statistics

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Summary

Given a hereditary graph property Ρ let Ρ n be the set of those graphs in Ρ on the vertex set {1, …, n}. Define the constant c n by \(\left| {P^n } \right| = 2^{cn(_2^n )} .\) We show that the limit limn → ∞ c n always exists and equals 1 − 1/r, where r is a positive integer which can be described explicitly in terms of Ρ. 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.

AMS Subject Classification: Primary 05C35, Secondary 05C30.