Stability theorems for cancellative hypergraphs

A cancellative hypergraph has no three edges A,B,C with AΔB⊂C. 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. the two forbidden subhypergraphs in a cancellative 3-graph is F5={abc,abd,cde}. For n⩾33 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 n⩾3000. th 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.