Hereditary and Monotone Properties of Graphs

Béla Bollobás () and Andrew Thomason ()
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Béla Bollobás: Trinity College
Andrew Thomason: DPMMS, Centre for Mathematical Sciences

A chapter in The Mathematics of Paul Erdős II, 2013, pp 69-80 from Springer

Abstract: 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.

Keywords: Monotonicity Property; Hereditary Properties; Forbidden Edges; Monochromatic Complete Subgraphs; Short Self-contained Proof (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4614-7254-4_6

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DOI: 10.1007/978-1-4614-7254-4_6

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