Generalized Turán densities in the hypercube

Abstract

A classical extremal, or Turán-type problem asks to determine ex(G,H), the largest number of edges in a subgraph of a graph G which does not contain a subgraph isomorphic to H. Alon and Shikhelman introduced the so-called generalized extremal number ex(G,T,H), defined to be the maximum number of subgraphs isomorphic to T in a subgraph of G that contains no subgraphs isomorphic to H. In this paper we investigate the case when G=Qn, the hypercube of dimension n, and T and H are smaller hypercubes or cycles.