Constructions of non-principal families in extremal hypergraph theory Original Research Article

A family image of image-graphs is called non-principal if its Turán density is strictly smaller than that of each individual member. For each image we find two (explicit) image-graphs image and image such that image 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.