Subhypergraph counts in extremal and random hypergraphs and the fractional <Emphasis Type="Italic">q</Emphasis>-independence

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.