Stability theorems for cancellative hypergraphs

A cancellative hypergraph has no three edges A;B;C with ADBCC: We give a new short proof of an old result of Bollobás, which states that the maximum size of a cancellative triple system is achieved by the balanced complete tripartite 3-graph. One of the two forbidden subhypergraphs in a cancellative 3-graph is F5 1⁄4 fabc; abd; cdeg: For nX33 we show that the maximum number of triples on n vertices containing no copy of F5 is also achieved by the balanced complete tripartite 3-graph. This strengthens a theorem of Frankl and Füredi, who proved it for nX3000: For both extremal results, we show that a 3-graph with almost as many edges as the extremal example is approximately tripartite. These stability theorems are analogous to the Simonovits stability theorem for graphs. r 2004 Elsevier Inc. All rights reserved.