A note on a conjecture by Füredi

In [Z. Füredi, Turán type problems, in: Surveys in Combinatorics, Guildford, 1991, in: London Math. Soc. Lecture Note Ser., vol. 166, Cambridge Univ. Press, Cambridge, 1991, pp. 253-300, MR1161467 (93d:05072)], Füredi raised a conjecture about the maximum size of L-intersecting families. In this note, we address a variant of this conjecture. In particular, we show that for any Steiner triple system S on [k], there exist a family F of k-sets on [n] with |F|=Ω(n2+?) and such that for every F₀∈F the family is isomorphic to S.