Constructions of non-principal families in extremal hypergraph theory

A family FF of kk-graphs is called non-principal   if its Turán density is strictly smaller than that of each individual member. For each k⩾3k⩾3 we find two (explicit) kk-graphs FF and GG such that {F,G}{F,G} is non-principal. Our proofs use stability results for hypergraphs. This completely settles the question posed by Mubayi and Rödl [On the Turán number of triple systems, J. Combin. Theory A, 100 (2002) 135–152].Also, we observe that the demonstrated non-principality phenomenon holds also with respect to the Ramsey–Turán density as well.