Stability theorems for cancellative hypergraphs

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Abstract

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. One of 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 F5is also achieved by the balanced complete tripartite 3-graph. This strengthens a theorem of Frankl and Füredi, who proved it for n≥3000.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. © 2004 Elsevier Inc.

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Keevash, P., & Mubayi, D. (2004). Stability theorems for cancellative hypergraphs. Journal of Combinatorial Theory. Series B, 92(1), 163–175. https://doi.org/10.1016/j.jctb.2004.05.003

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