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Subhypergraph counts in extremal and random hypergraphs and the fractional q-independence

Author

Listed:
  • Andrzej Dudek

    (Emory University)

  • Joanna Polcyn

    (Adam Mickiewicz University)

  • Andrzej Ruciński

    (Adam Mickiewicz University)

Abstract

We study the extremal parameter N(n,m,H) which is the largest number of copies of a hypergraph H that can be formed of at most n vertices and m edges. Generalizing previous work of Alon (Isr. J. Math. 38:116–130, 1981), Friedgut and Kahn (Isr. J. Math. 105:251–256, 1998) and Janson, Oleszkiewicz and the third author (Isr. J. Math. 142:61–92, 2004), we obtain an asymptotic formula for N(n,m,H) which is strongly related to the solution α q (H) of a linear programming problem, called here the fractional q-independence number of H. We observe that α q (H) is a piecewise linear function of q and determine it explicitly for some ranges of q and some classes of H. As an application, we derive exponential bounds on the upper tail of the distribution of the number of copies of H in a random hypergraph.

Suggested Citation

  • Andrzej Dudek & Joanna Polcyn & Andrzej Ruciński, 2010. "Subhypergraph counts in extremal and random hypergraphs and the fractional q-independence," Journal of Combinatorial Optimization, Springer, vol. 19(2), pages 184-199, February.
    • Handle: RePEc:spr:jcomop:v:19:y:2010:i:2:d:10.1007_s10878-008-9174-9
    DOI: 10.1007/s10878-008-9174-9
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