For a family of r-graphs F, the Turan 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 Turan density graphics When F is an r-graph, pi(F) not equal 0, and r > 2, determining pi(F) is a notoriously hard problem, even for very simple r-graphs F. For example, when r = 3. the value of pi(F) is known for very few (< 10) irreducible r-graphs. Building upon a method developed recently by de Caen and Furedi (J. Combin. Theory Ser. B 78 (2000), 274276), we determine the Turan 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 Furedi (Combinatorica 3 (1983), 341-349) that pi(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, 1241 234, and let 165 be the 3-graph 123, 234, 345 7 451, 512. We prove circle (4)/(9) less than or equal to pi (F (3, 2)) less than or equal to (1)/(2) circle pi ({K-4(-),C-5}) less than or equal to (10)/(31) = 0.322581, circle 0.464 < pi(C-5) less than or equal to 2 - root2 < 0.586. [References: 12]
展开▼
