On the Turán Number of Triple Systems

Abstract

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 )≠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. B 78 (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 )= 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 C 5 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 Füredi ( Discrete Math. 50 (1984) 323–328) that π( K − 4)= 2 7 . The best known bounds are 2 7 ⩽ π( K − 4)⩽ 1 3 .