Stability theorems for some Kruskal-Katona type results

Xizhi Liu, Sayan Mukherjee

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, 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.