On the Turلn Number of Triple Systems

For a family of r-graphs F the Turán 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 Turán density π(F)=limn→∞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 Füredi (J. Combin. Theory Ser. B78 (2000), 274–276), we determine the Turán 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 Füredi (Combinatorica 3 (1983), 341–349) that π(H)=29, 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 &#x02022; 2))⩽12, x02022; })⩽=0.322581, x02022; lt;(C)⩽2−√2<0.586. iddle result is related to a conjecture of Frankl and Füredi (Discrete Math.50 (1984) 323–328) that π(K−4)=27. The best known bounds are 27⩽π(K−4)⩽13.