On the Turán 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) = lim n→∞ ex(n,F) ( n r ) . When F is an r-graph, π(F) 6= 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 [3], 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 [5] 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 • 4/9 ≤ π(F(3, 2)) ≤ 1/2, • π({K− 4 , C5}) ≤ 10/31 = 0.322581, • 0.464 < π(C5) ≤ 2− √ 2 < 0.586. The middle result is related to a conjecture of Frankl and Füredi [6] that π(K− 4 ) = 2/7. The best known bounds are 2/7 ≤ π(K− 4 ) ≤ 1/3. ∗School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA; research supported in part by the National Science Foundation under grant DMS-9970325 †Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA; research supported in part by the National Science Foundation under grant DMS-0071261 1991 Mathematics Subject Classification: 05C35, 05C65, 05D05