Book contents
- Frontmatter
- Contents
- Preface
- List of contributors
- Acknowlegements
- Generating expanders from two permutations
- Sum-free subsets
- Is there a different proof of the Erdős-Rado theorem?
- Almost collinear triples among N points on the plane
- Hamilton cycles in random graphs of minimal degree at least k
- The circumference of a graph with a given minimal degree
- On arithmetic progressions in sums of sets of integers
- On graphs not containing prescribed induced subgraphs
- Partitions sans petits sommants
- A compact sequential space
- The critical parameter for connectedness of some random graphs
- Multiplicative functions on arithmetic progressions: III. The large moduli
- Locally finite groups of permutations of ℕ acting on l∞
- Hypergraph games and the chromatic number
- On arithmetic graphs associated with integral domains
- On the number of certain subgraphs of graphs without large cliques and independent subsets
- Sets of multiples of Behrend sequences
- A functional equation arising from mortality tables
- The differences between consecutive primes, IV
- On the cofinality of countable products of cardinal numbers
- On σ-centered posets
- A Galvin–Hajnal conjecture on uncountably chromatic graphs
- Necessary conditions for mean convergence of Hermite–Fejér interpolation
- On the Erdős–Fuchs theorems
- A tournament which is not finitely representable
- On the volume of the spheres covered by a random walk
- Special Lucas sequences, including the Fibonacci sequence, modulo a prime
- A remark on heights of subspaces
- Incompactness for chromatic numbers of graphs
- Graphs with no unfriendly partitions
- On the greatest prime factor of an arithmetical progression
- The probabilistic lens: Sperner, Turan and Bregman revisited
- On the mean convergence of derivatives of Lagrange interpolation
- Sur une question d'Erdős et Schinzel
- Large α-preserving sets in infinite α-connected graphs
- Some recent results on interpolation
- Partitioning the quadruples of topological spaces
- Some of my favourite unsolved problems
On graphs not containing prescribed induced subgraphs
Published online by Cambridge University Press: 05 March 2012
- Frontmatter
- Contents
- Preface
- List of contributors
- Acknowlegements
- Generating expanders from two permutations
- Sum-free subsets
- Is there a different proof of the Erdős-Rado theorem?
- Almost collinear triples among N points on the plane
- Hamilton cycles in random graphs of minimal degree at least k
- The circumference of a graph with a given minimal degree
- On arithmetic progressions in sums of sets of integers
- On graphs not containing prescribed induced subgraphs
- Partitions sans petits sommants
- A compact sequential space
- The critical parameter for connectedness of some random graphs
- Multiplicative functions on arithmetic progressions: III. The large moduli
- Locally finite groups of permutations of ℕ acting on l∞
- Hypergraph games and the chromatic number
- On arithmetic graphs associated with integral domains
- On the number of certain subgraphs of graphs without large cliques and independent subsets
- Sets of multiples of Behrend sequences
- A functional equation arising from mortality tables
- The differences between consecutive primes, IV
- On the cofinality of countable products of cardinal numbers
- On σ-centered posets
- A Galvin–Hajnal conjecture on uncountably chromatic graphs
- Necessary conditions for mean convergence of Hermite–Fejér interpolation
- On the Erdős–Fuchs theorems
- A tournament which is not finitely representable
- On the volume of the spheres covered by a random walk
- Special Lucas sequences, including the Fibonacci sequence, modulo a prime
- A remark on heights of subspaces
- Incompactness for chromatic numbers of graphs
- Graphs with no unfriendly partitions
- On the greatest prime factor of an arithmetical progression
- The probabilistic lens: Sperner, Turan and Bregman revisited
- On the mean convergence of derivatives of Lagrange interpolation
- Sur une question d'Erdős et Schinzel
- Large α-preserving sets in infinite α-connected graphs
- Some recent results on interpolation
- Partitioning the quadruples of topological spaces
- Some of my favourite unsolved problems
Summary
Introduction
Given a fixed graph H on t vertices, a typical graph G on n vertices contains many induced subgraphs isomorphic to H as n becomes large. Indeed, for the usual model of a random graph G* on n vertices (see [4]), in which potential edges are independently included or not each with probability ½, almost all such G* contain induced copies of H as n → ∞. Thus, if a large graph G contains no induced copy of H, it deviates from being ‘typical’ in a rather strong way. In this case, we would expect it to behave quite differently from random graphs in many other ways as well. That this in fact must happen follows from recent work of several authors, e.g., see Chung, Graham & Wilson [5] and Thomason [7], [8]. In this paper we initiate a quantitative study of how various deviations of randomness are related. The particular property we investigate (‘uniform edge density for half sets’ – see Section 3) is just one of many which might have been selected and for which the same kind of analysis could be carried out.
This work also shares a common philosophy with several recent papers of Alon & Bollobás [1] and Erdős & Hajnal [6], which investigate the structure of graphs which have an unusually small number of non-isomorphic induced subgraphs. This is a strong restriction and such graphs must have very large subgraphs which are (nearly) complete or independent.
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- A Tribute to Paul Erdos , pp. 111 - 120Publisher: Cambridge University PressPrint publication year: 1990
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