Arkiv för Matematik

Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs

Svante Janson and Andrzej Ruciński

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General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph ℋ, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.


A. Ruciński supported by Polish grant N201036 32/2546. Research was performed while the authors visited Institut Mittag-Leffler in Djursholm, Sweden, during the program Discrete Probability, 2009.

Article information

Ark. Mat., Volume 49, Number 1 (2011), 79-96.

Received: 8 May 2009
First available in Project Euclid: 31 January 2017

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2010 © Institut Mittag-Leffler


Janson, Svante; Ruciński, Andrzej. Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs. Ark. Mat. 49 (2011), no. 1, 79--96. doi:10.1007/s11512-009-0117-1.

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  • Dudek, A., Polcyn, J. and Ruciński, A., Subhypergraph counts in extremal and random hypergraphs and the fractional q-independence, J. Comb. Optim. 19 (2010), 184–199.
  • Graham, R., Rödl, V. and Ruciński, A., On Schur properties of random subsets of integers, J. Number Theory 61 (1996), 388–408.
  • Janson, S., Oleszkiewicz, K. and Ruciński, A., Upper tails for subgraph counts in random graphs, Israel J. Math. 142 (2004), 61–92.
  • Janson, S., Łuczak, T. and Ruciński, A., Random Graphs, Wiley, New York, 2000.
  • Janson, S. and Ruciński, A., The infamous upper tail, Random Structures Algorithms 20 (2002), 317–342.
  • Janson, S. and Ruciński, A., The deletion method for upper tail estimates, Combinatorica 24 (2004), 615–640.
  • Łuczak, T. and Prałat, P., Chasing robbers on random graphs: zigzag theorem, Preprint, 2008.
  • Rödl, V. and Ruciński, A., Rado partition theorem for random subsets of integers, Proc. Lond. Math. Soc. 74 (1997), 481–502.
  • Vu, V. H., A large deviation result on the number of small subgraphs of a random graph, Combin. Probab. Comput. 10 (2001), 79–94.