Constructions of Tur\'an systems that are tight up to a multiplicative constant

For positive integers n s> r, the Tur\'an function T (n, s, r) is the smallest size of an r-graph with n vertices such that every set of s vertices contains at least one edge. Also, define the Tur\'an density t (s, r) as the limit of T (n, s, r) / n r as n. The question of estimating these parameters received a lot of attention after it was first raised by Tur\'an in 1941. A trivial lower bound is t (s, r) 1/s s-r. In the early 1990s, de Caen conjectured that r t (r+1, r) as r. We disprove this conjecture by showing more strongly that for every integer R1 there is R (in fact, R can be taken to grow as (1+o (1) ) \, R R) such that t (r+R, r) (R+o (1) ) / r+R R as r, that is, the trivial lower bound is tight for every R up to a multiplicative constant R.