### Abstract

Random graphs are graphs with a fixed number of vertices, where each edge is present with a fixed probability. We are interested in the probability that a random graph contains a certain pattern, for example a cycle or a clique. A very high edge probability gives rise to perhaps too many edges (which degrades performance for many algorithms), whereas a low edge probability might result in a disconnected graph. We prove a theorem about a threshold probability such that a higher edge probability will asymptotically almost surely produce a random graph with the desired subgraph.

### License

BSD License

### Topics

- Mathematics/Graph theory
- Mathematics/Probability theory

### Session Random_Graph_Subgraph_Threshold

- Ugraph_Misc
- Prob_Lemmas
- Ugraph_Lemmas
- Ugraph_Properties
- Subgraph_Threshold

### Depends on

- A Probabilistic Proof of the Girth-Chromatic Number Theorem

### Used by

- Turán’s Graph Theorem
- The Balog–Szemerédi–Gowers Theorem
- Roth’s Theorem on Arithmetic Progressions

## Cite

×

Random_Graph_Subgraph_Threshold-AFP

    @article{Random_Graph_Subgraph_Threshold-AFP,
      author  = {Lars Hupel},
      title   = {Properties of Random Graphs -- Subgraph Containment},
      journal = {Archive of Formal Proofs},
      month   = {February},
      year    = {2014},
      note    = {\url{https://isa-afp.org/entries/Random_Graph_Subgraph_Threshold.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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