Subhypergraph counts in extremal and random hypergraphs and the fractional q-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 alpha (q) (H) of a linear programming problem, called here the fractional q-independence number of H. We observe that alpha (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.
hypergraphs random hypergraphs fractional independence