Stability theorems for some Kruskal–Katona type results

Abstract

The classical Kruskal–Katona theorem gives a tight upper bound for the size of an r-uniform hypergraph H as a function of the size of its shadow. Its stability version was obtained by Keevash who proved that if the size of H is close to the maximum with respect to the size of its shadow, then H is structurally close to a complete r-uniform hypergraph. We prove similar stability results for two classes of hypergraphs whose extremal properties have been investigated by many researchers: the cancellative hypergraphs and hypergraphs without expansion of cliques.