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On the number of copies of one hypergraph in another

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Abstract

Given two hypergraphsH andG, letN(G, H) denote the number of subhypergraphs ofG isomorphic toH. LetN(l, H) denote the maximum ofN(G, H), taken over allG with exactlyl edges. In [1] Noga Alon analyzes the asymptotic behaviour ofN(l, H) forH a graph. We generalize this to hypergraphs:

Theorem:For a hypergraph H with fractional cover number ρ*,N(G,H).=θ(lρ*)

The interesting part of this is the upper bound, which is shown to be a simple consequence of an entropy lemma of J. Shearer.

In a special case which includes graphs, we also provide a different proof using a hypercontractive estimate.

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Ehud Friedgut

  2. Department of Mathematics and RUTCOR, Rutgers University, 08903, New Brunswick, NJ, USA

    Jeff Kahn

Authors
  1. Ehud Friedgut
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  2. Jeff Kahn
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Correspondence to Ehud Friedgut.

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Research supported in part by NSF.

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Friedgut, E., Kahn, J. On the number of copies of one hypergraph in another. Isr. J. Math. 105, 251–256 (1998). https://doi.org/10.1007/BF02780332

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  • DOI: https://doi.org/10.1007/BF02780332

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