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 ⩽(F(3,2))⩽1/2,
({:K,C})⩽=0.322581,
0.464<(C)⩽2−√2<0.586.
The middle result is related to a conjecture of Frankl and Fu¨redi (Discrete Math.50 (1984) 323–328) that π(K−4)=2/7. The best known bounds are 2/7⩽π(K−4)⩽1/3 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 ⩽(F(3,2))⩽1/2,
({:K,C})⩽=0.322581,
0.464<(C)⩽2−√2<0.586.
The middle result is related to a conjecture of Frankl and Fu¨redi (Discrete Math.50 (1984) 323–328) that π(K−4)=2/7. The best known bounds are 2/7⩽π(K−4)⩽1/3, 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 ⩽(F(3,2))⩽1/2,
({:K,C})⩽=0.322581,
0.464<(C)⩽2−√2<0.586.
The middle result is related to a conjecture of Frankl and Fu¨redi (Discrete Math.50 (1984) 323–328) that π(K−4)=2/7. The best known bounds are 2/7⩽π(K−4)⩽1/3,({:K,C})⩽=0.322581,0.464<(C)⩽2−√2<0.586. The middle result is related to a conjecture of Frankl and Fu¨redi (Discrete Math.50 (1984) 323–328) that π(K−4)=2/7. The best known bounds are 2/7⩽π(K−4)⩽1/3. The best known bounds are 2/7⩽π(K−4)⩽1/3⩽π(K−4)⩽1/3. [Copyright &y& Elsevier]