Generalized Turán densities in the hypercube

Abstract

A classical extremal, or Turán-type problem asks to determine $\mathrm{ex}(G,H)$, 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 $\mathrm{ex}(G,T,H)$, 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 $G=Q_n$, the hypercube of dimension $n$, and $T$ and $H$ are smaller hypercubes or cycles.