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.
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The research of A. Ruciński was supported by the Polish grant N201036 32/2546. Part of this research was done while on leave at Emory University.
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Dudek, A., Polcyn, J. & Ruciński, A. Subhypergraph counts in extremal and random hypergraphs and the fractional q-independence. J Comb Optim 19, 184–199 (2010). https://doi.org/10.1007/s10878-008-9174-9
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DOI: https://doi.org/10.1007/s10878-008-9174-9