Explicit constructions of triple systems for Ramsey-Turán problems

We explicitly construct four infinite families of irreducible triple systems with Ramsey-Turán density less than the Turán density. Two of our families generalize isolated examples of Sidorenko [14], and the first author and Rödl [12]. Our constructions also yield two infinite families of irreducible triple systems whose Ramsey-Turán densities are exactly determined. For an r-graph F, the Turán number ex(n, F) is the maximum number of edges in an n vertex exists, but r-graph containing no copy of F. It is well known that π(F) = limn→ ∞ ex(n, F) / � n r these numbers are very hard to determine when r ≥ 3. For example, until very recently [10] no nontrivial infinite family {Fi} of triple systems has been constructed for which π(Fi) is known (by “nontrivial ” we mean that for every Fi, there are no two vertices x, y of Fi for which (1) no edge contains both x and y, and (2) xuv is an edge iff yuv is an edge). Two examples that are known, and used in this note, are π(F5) = 2/9, and π(F (3, 2)) = 4/9, where F5 = {123, 124, 345} an