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 ( n 2 + ϵ ) and such that for every F 0 ∈ F the family { F ∩ F 0 | F ∈ F , F ≠ F 0 } is isomorphic to S .