  There is much recent interest in understanding the density at which constant
size graphs can appear in a very large graph. Specifically, the inducibility of
a graph H is its extremal density, as an induced subgraph of G, where |G| ->
infinity. Already for 4-vertex graphs many questions are still open. Thus, the
inducibility of the 4-path was addressed in a construction of Exoo (1986), but
remains unknown. Refuting a conjecture of Erdos, Thomason (1997) constructed
graphs with a small density of both 4-cliques and 4-anticliques. In this note,
we merge these two approaches and construct better graphs for both problems.
