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We construct a Ramsey class whose objects are Steiner systems. In contrast to the situation with general r -uniform hypergraphs, it turns out that simply putting linear orders on their sets of vertices is not enough for this purpose: one also has to strengthen the notion of subobjects used from “induced subsystems” to something we call “strongly induced subsystems”. Moreover we study the Ramsey properties of other classes of Steiner systems obtained from this class by either forgetting the order or by working with the usual notion of subsystems. This leads to a perhaps surprising induced Ramsey theorem in which designs get coloured. [ABSTRACT FROM AUTHOR]

The Erdo˝s–Stone–Simonovits Theorem implies that the Tura´n density of a family of graphs is the minimum of the Tura´n densities of the individual graphs from the family. It was conjectured by Mubayi and Ro¨dl (J. Combin. Theory Ser. A, submitted) that this is not necessarily true for hypergraphs, in particular for triple systems. We give an example, which shows that their conjecture is true. [Copyright &y& Elsevier]

For a family of r-graphs F the Tura´n number ex (n, F) is the maximum number of edges in an n vertex r-graph that does not contain any member of F. The Tura´n density π(F)=limlower limit n→∞ ex(n,F)/((nr). When F is an r-graph, π(F)≠0, and r>2, determining π(F) is a notoriously hard problem, even for very simple r-graphs F. For example, when r=3, the value of π(F) is known for very few (<10) irreducible r-graphs. Building upon a method developed recently by de Caen and Fu¨redi (J. Combin. Theory Ser. B78 (2000), 274–276), we determine the Tura´n densities of several 3-graphs that were not previously known. Using this method, we also give a new proof of a result of Frankl and Fu¨redi (Combinatorica 3 (1983), 341–349) that π(H)=2/9, where H has edges 123,124,345. Let F(3,2) be the 3-graph 123,145,245,345, let K−4 be the 3-graph 123,124,234, and let C5 be the 3-graph 123,234,345,451,512. We prove